Science fiction sometimes plays with the concept of a planet with different geological structure than earth (for instance traveling through liquid water core in Star Wars) yet such imaginations often lack scientific plausibility.
What is scientifically plausible and what is not then? Are there any serious research (or at least data-justified scientific speculation) about geology of exoplanets and how it differs from earth?
Some of the main features to consider when you want to think about the geology (not the best word here) of a planet are:
Does it have a gyro, like a molten core of nickel=iron. This creates poles (magnetism) and with imagination that can lead to lots of creative possibilities.
Is there water
Is there an atmosphere, and what kind?
What is the temperature? (This overlaps the above two, and is also affected by proximity to its star.)
Are there moons. This impacts tides.
Is it a gas giant? A liquid planet is possible. Some of our gas giants are essentially liquid. It sounds like this is what you are reaching for.
As far as elemental makeup, I would be surprised if we found areas of space with wildly different compositions. "We are all stardust" is an expression based on the idea that the elements in our solar system were formed in stars, and so the smallest elements are most abundant. This should hold elsewhere.
Once you've considered those 6 ideas, you can start to generate a lot of different combinations.
Here is some hard science to help you out:
This abstract should start you off. There is quite a lot of research to back up what we know about planets and exoplanets including the bodies in our solar system.
An additional link, with models and equations.
A third link that describes our solar system planets.
Tip: Go to google scholar and type 'geology of exoplanets." You will find more articles than you can shake a stick at.
With (considerably) more specificity in your question, an equation or two in this answer could make sense. Just ping me if that's what you are actually looking for! (I believe what you are asking is 'how crazy of a planet can exist, and be scientifically valid,' not 'how do I use equations to derive the probability of a liquid planet, or some such.') You might also want to consider moons. They are becoming more and more en vogue in planetary science.
"What is scientifically plausible and what is not then?" - Since science is not a set of dogmas, what is scientifically plausible today is not what is going to be scientifcally plausible tomorrow. Verne wrote about a hollow earth; we now know this isn't possible, but it was plausible in his day. If his story stood exclusively in its scientific plausibility, it would be unreadable now. Since it is still readable, it follows that it stands in something else - plot, characters, writing (not something big with Verne either), fantasy, etc.
Writing something that is scientifically plausible today is to write something that won't be scientifically plausible in twenty years. Short of trying to guess what is going to be scientifically plausible in a few decades, your best chance is to rely on other characteristics of your story. Or to pretend to guess what is going to be scientifically plausible in a thousand years. This is, of course, phlebotinology or handwaviumlogy; it may make you a visionary prophet long after your death, if by chance you get something right, but it won't make you a great writer or even a plausible one during your life, if you can't can come up with a story that can stand to literary scrutiny once its supposed scientific plausibility wanes off, or albeit never having had it (Asimov's golden eggs bird, anyone?)
So, what is (scientifically) plausible today?
Planets will be spheroids, with the equatorial diameter slightly bigger than the meridional diameter (how much, it depends on rotation and composition).
Planets will be chiefly composed of common elements, easily starbaked according to their nuclear physics, or common products of nuclear decay/collision: hydrogen, helium, carbon, oxygen, nitrogen, silicium, iron, with smaller amounts of other elements with an atomic number below 26, and even smaller amounts of elements beyond 26. Higher atomic numbers will imply newer, second generation stars, already formed from the debris of older, collapsed stars.
Planets will mainly be, chemically speaking, of one of three kinds: 1. gas giants (like Jupiter, Saturn, Neptune, Uranus), which consist in enormous balls of gasses - hydrogen, helium, methane, water, carbonic dioxide, nitrogen. Free oxygen will be rare because with so much hydrogen and carbon, most of it will combine with those elements. They need to be big or their gravity won't be able to hold those gasses. And albeit the name, they will be mostly solid - solid hydrogen, methane, carbonic dioxide, compressed into solid state by enormous pressures. 2. rockballs (like Mars, Venus, our Moon, and the formerly-known-as-a-planet Pluto), smaller spheres of oxygen-silicium-aluminum-carbon solid composites. 3. terrestrial planets (like Earth and Mercury), also relatively small spheres with an iron-nickel core covered by a layer of oxygen-silicium-carbon composites. Types 2 and 3 need to be small, otherwise their accordingly higher gravity will retain so much lighter gasses that they will in time become gas giants. Type 3 will probably have a magnetic field that shields them from radiation from their stars. Terrestrial planets and rockballs may (like Earth or Venus) or may not (like Mercury and Mars) have atmospheres; gas giants arguably are their atmospheres. How much atmosphere a planet has, and which gasses compose it, depends on its gravity, which in turn is a function of their mass and density: smaller planets will have little to no atmosphere, and more likely will retain heavier gasses; bigger planets will retain even hydrogen.
Planets, regardless of type and size, will have a solid nucleus, because at the pressure levels implied by their size (and this includes dwarf planets like Pluto), any substance, even helium, will be solid.
Planets will orbit stars (well, there may be "rogue planets", but then they really don't match the standing definition of planet, which requires an orbit), and their existence is dependent on those stars. Planets around blue giants will have a "short" (in astronomic terms) existence. Planets around red dwarves will last longer. Our own sun is a yellow G-type star, much loger-lived than giants, but still doomed to catastrophic extinction in the future.
Planets distance from their star, along with star type, is key for their temperature, which is in turn key to what elements will be present in solid/liquid/gaseous state. Too far away, almost everything will be solid; too close, almost everything in their surface will be gaseous. How far is "too far" or "too close" depends on star type; "too far" from a red dwarf is quite certainly "too close" to a blue giant. In our own system, Mercury and Venus are too close, and Mars is borderline too far. "Rogue planets" by definition are "too far" from any star.
Planets may or may not have moons, which, if they are big enough relatively to the planet, may in turn affect their sismology (and oceanography, if they have partially liquid surfaces). In our system, only Earth and ex-planet Pluto have companions that can significantly affect those aspects; Venus and Mercury have no moons, Mars has only very small ones, and the gas giants are too big to be significantly affected by their moons, even when these are as huge as Earth.
Planets that have atmospheres will have different atmospheres depending on their temperature, composition, and the presence of life. Oxygenated atmospheres are unlikely unless they harbour life; otherwise oxygen will combine with carbon and hydrogen, and excess carbon and hydrogen will likely be combined as methane.
Some components of an atmosphere (methane, carbon dioxide) will have a greenhouse effect and increase temperature. Some planets, like Venus, will be dominated by this; others, like Earth, not so much, as long as we do not burn too much oil into their atmosphere.
Planets rotate; if their rotation period is the same as their orbital period, we say that they are tide locked; only half of their surface is ever exposed to sunlight, and the other half is permanently dark (except, of course, for what light may be provided by other stars and by eventual satellites). This will make their temperature much higher on one side and much lower on the other, regardless of the distance from the star. The rotation speed is limited, however; if it is too fast, regarding its composition, the planet will disintegrate.
Planets will have some tilt - a difference between the plane of their orbits and the plane of their rotations. Earth has a 23 degree-ish tilt, and it causes our seasons, as well as significantly longer days or nights in higher latitudes. Planets with huge absolute tilts (like Uranus) will have more intense and complicated seasons, and the tilt may even interfere with the duration of their days and nights even in their equatorial line; planets with small tilts (like Mercury) will have little to no seasons.
Other characteristics that are not present in our system may be more or less frequent: elongated orbits (which may create season-like phenomena independent of tilt, or combine with tilt to create complex seasonal effects), multiple stars, variable stars, non-system stars much closer to the planet than we are used to in Earth (like in a galactical core), single-planet systems, systems in which gas giants are closer to the star than rockballs and terrestrials, gas-giant-only systems.
Most of the above applies for satellites (except the relation between tidal lock and perpetual day/night zones). Additionally, the orbit of satellites around their planets may have additional effects on their temperatures (and they will probably experience much more notable solar eclypses if their orbital planet is similar to their planets' orbital plane).
Pretty much everything else will be "implausible" from a strictly 2017-scientific standpoint: planets with weird compositions such as a planet with an osmium core or more generally with any huge proportions of heavier-than-iron elements, planets with more chlorine than oxygen or more lithium than iron, planets without any silicium, all-water (all cheese, etc) planets, hollow planets, toroidal (cubic, icosaedral, ellipsoid, flat, anything non-spheroidal) planets, non-elliptical orbits, anti-matter planets orbiting normal-matter stars or conversely, dark-matter planets, non-matter planets, planets with short precessions, planets that have three-minute orbits, planets that oribt a blue giant but are several billion years old, smallish planets with an atmosphere, planets that have magnetic fields without an iron-nickel core, planets that deflect radiation without a magnetic field, etc, etc, etc. Those would require magic or pseudo-scientific justifications, appeals to the future, or direct suspension of disbelief.
