Intro
There are a lot of physical issues with this planet being connected by the end of a bean-stock that is also connected to Earth, but if its magical anything is possible.
I'm going to first examine the composition of our new planet and how we might approach the minimization of its radius.
Earth-Like Gravity
The most obvious issue for minimizing the radius of this planet is mass, since I'm assuming you want the gravity to be at least somewhat similar to Earth's. For the smallest dimensions of a Earth like planet that are physically possible the density will be the most important parameter, which of course constrains composition.
Now there are a few issues that really constrain the possible solutions to the problem. First if the core has to have the same diameter as that of Earth and has to have the same iron based composition, then we'll start there. The core is composed of an inner and outer component, so I'll assume the core in it's entirety.
From my handwavey calculations I guestimated the mass to be about $8x10^{23}$ kg, from researching it a bit I found the core is estimated to be about $1.9 x 10^{24}$ kg which is about a third of the Earth's total mass. [O.G. Sorokhtin et. al, Dev. in E&ES, 2011] I'll use this later figure.
The total radius so far is about 3500 km. We need to account for a remaining $4x10^{24}$ kg of mass. We want a dense material, but it must not be denser than the core. If we add another 1850-2000 km for the mantel we can get an average density between 7.7-8.7 $g/cm^{3}$. This is not unreasonable for a metallic composition. However, this is another issue altogether; metal is way too thermally conductive to work if the core is to be the same as Earth's. A particularly cold core wouldn't actually affect the surface temperature (after all this is determined atmospherically and not from the internal temperature). However, a cold core would result in a weak or lack of internal dynamo effect and no magnetic field, which of course make life impossible. To resolve this, you could force it to work, of course the solution is completely contrived and result in some rather bizarre things happening; a mix of heavy metalloids and oxygen might do the trick.
So now we have a reasonably dense planet with the same gravity as Earth. Coming out with a planet about 5500 km including the crust, our planet will still have near Earth gravity while being a bit smallar. However, our planet is still much larger than Mars. The circumference of this planet would just be a few thousand km smaller than Earth. While this is quite a bit, it probably wouldn't feel too much different when on the surface. The curvature of this planet would be noticeable at lower altitudes, but a few thousand kilometers isn't as much as it sounds when we consider that the the Earth's circumference is 40,000 km! If someone is in the US for example, any place within the continental US is no more than 5000 km away.
Other options
Now, you mentioned a lower gravity. But the question is how much lower do you want to go. The lower the gravity, the more bizarre the inhabitants of this planet. I'm not much of a biologist, but I'm aware of the basic effects of lower gravity might have upon organisms. If the sort of animals and plants on this planet were similar to the ones on Earth, they would probably tend to be taller. They probably would be more massive as well since the limit to being crushed under ones' own weight would increase. But the animals wouldn't be the only problem here.
Lowering the gravity allows for less extra planetary mass as well and possibly less gravitational acceleration. But we'll still need a sizable mantle for a variety of reasons.
If we just isolate the Core, we actually get more gravitational acceleration than on the surface of the Earth: about 11 $m/s^{2}$ compared to our 9.81 $m/s^{2}$. A fairly thick mantle will still be necessary or else the mantle will be crushed under the gravitational attraction as well as being melted due to the heat transfer. There's also the problem of the core losing all the heat and another problem of no magnetic field as described above. Furthermore the outer core is fluid (the inner core is solid) and the Earth's dynamo is primarily driven by the outer core, with convection cycles running up into the lower mantel.
It is often tempting to cobble together basic calculations using hand-wavy equations for heat transfer, perhaps using the density of a candidate composition to estimate how thick our mantel might be, maybe we might go from there to solve the gravitational acceleration. This unfortunately would result in patent nonsense no better than just making something up which sounded nice. To properly get a sense of the interactions which go into these calculations we would need to create an equation of state for this planet using the interplay among pressure, heat and temperature. From this we can get a sense of how the pressure and density interact and how heat would transfer for various materials which are allowed under the density/pressure constrains. To properly do these calculations would require an entire thesis unto itself.
If you are interested in seeing an example of the sort of calculations required, see
[http://articles.adsabs.harvard.edu//full/1980LPSC...11.1999A/0001999.000.html]
which contains the article "Equations of State in Planet interiors" by Orson Anderson and John Baumgardner.
All in all, insisting that the core of the bean-stock linked planet must be the same as Earth's core, in both composition and heat, seriously restricts the possibiities for how realistic the size can be. For the resulting planet to be realistically suitable to an Earth surface, it will end up having a size not to far from Earth's current size. As we've explored, keeping the Earth like gravity will result in a very Earth sized planet. Allowing the gravity to be less than Earth's will decrease circumference a bit, but I would not suspect the results to be too extreme.
The interesting thing about systems such as these, is that the interplay found in the EoS really restricts the possibilities. I'd estimate that at the smallest, you'd end up with a planet not too differently sized that the one described above with Earth like gravity.
Small Earth-like planet with low gravity
A possibility for envisioning an Earth-like planet with low gravity would be to have a much smaller core. The pressure of Earth's core is still too low for iron/nickel to be created from pressure alone. As regarding the dynamo effect, many materials can produce dynamo effects in theory, Iron is not special at core temperatures as it loses its ferromagnetic properties above the Curie temperature which is well below the temperature in Earth's core. So there are lots of choices for materials. Mars is a great example, as it is about half of Earths radius, but about a tenth of Earth's mass. If the radius were even smaller given it's mass, the gravity, which is a little over a third of Earth's, would increase.