I'm imagining an Earth-like world with a single large continent centered on the north pole, and not reaching entirely to the equator. As ocean currents play a large role in shaping weather, I'm curious how the lack of landmass at the equator might change climate zones on my planet. Would we still find tropical, subtropical, temperate, and tundra climates at similar latitudes as we would on Earth?
As noted I'm leaving my original answer to the rather vague original question right where it is because I think it's the right answer for that question; what you're asking now is a very different question.
What you're talking about is the weather and ocean cycles around a polar "super-continent" I don't have the expertise to give a truly comprehensive answer to this, I don't think anyone does, but there's a few things that we do know from the fossil record of Pangaea:
The oceans will be poorly oxygenated due to a few factors 1. their average depth is massively increased over the modern era, unless your world has much shallower ocean basins the same will be true there too 2. the shape of a single huge ocean basin does not encourage wide circulation, so instead of the Thermohaline Global Conveyor you get something more like Hadley Cells in the surface waters and no deep mixing 3. the atmosphere is higher in Carbon Dioxide, for reasons that will be discussed below, so the surface waters are getting less oxygen anyway 4. the warm tropical waters are going to have a lower maximum oxygen saturation as well. So there's not going to be much life in the oceans, which isn't to say none just not the volume and variety we're used to.
Sluggish ocean circulation is going to increase the equator-pole temperature differential, equatorial waters are going to be hotter than on Earth and the poles cooler. There will still be atmospheric circulation but it moves less heat.
Tropical waters are still going to spawn cyclonic storms (call them hurricanes, cyclones, or typhoons depending where you're from) and they're going to be catastrophic. There are two factors here the first is temperature, instead of a storm season when the surface waters are hot enough to produce cyclones (37 Degree Celsius surface water if memory serves) the surface waters, right at the equator at least, will be hot enough to spawn a constant stream of these storms. To make matters worse instead travelling across a few hundred kilometers and then making landfall, and being cut off from heat and water vapour, these storms can now stay over hot waters that feed them for thousands, if not tens of thousands, of kilometers growing all the time. The modern Cat 5 Hurricane is going to be a small tropical storm on this world.
The centre of continents are more prone to temperature extremes than coastal environments, this is the same on super-continents, the interior tends to be hot/cold and dry and pretty bare. In this scenario it will be much worse because of where the continent is, the interior will be permanently frozen even in high summer and any water vapour that penetrates will precipitate out immediately as ice in one form or another. There will be only thin coastal strips where there's any regular rainfall and that will probably be unpredictable and often violent.
Lack of erosion is going to be a problem, erosion, especially of the Himalaya, strips a lot (a gigaton or so every year) of Carbon Dioxide out of the modern atmosphere but super continents don't have large mountain ranges exposed to continuous weathering, the major subduction driven mountain ranges are deep in the interior, in this case frozen under a mile or so of ice.
There's likely to be a biome progression from the southern coast back toward the northern interior with lush storm battered coastal temperate rain forests giving way to drier shrublands, then vast grassland steppes and finally a vast sparse tundra right up to the edge of the ice fields. These biomes will be as much, or more, driven by the availability of moisture as by the annual average temperature or seasonal frost nights etc...
Some notes: the term "Tropical" is latitude defined and changes depending on rotation axis tilt, it's the latitude at the which the noon sun is directly overhead at mid summer, any farther from the equator the sun is never directly overhead at noon but further down the sky as your latitude increases. I've assumed you were still going with the southern coast around the 45 degrees latitude mark of the original question when looking at the land biomes you would and wouldn't see.
To answer that last question, you'd see the tropical and polar average temperature/climate bands expand due to the relative lack of heat circulation with the subtropical and temperate zones getting squeezed thin between them, all the temperature zones would be more linear in their distribution since you don't have the warming or cooling of say the Gulf Stream in Europe or the Humboldt Current along the Chilean coast. The driving force in vegetation cover types and distribution will be water availability rather than temperature and I would expect that to be in direct relation to the distance from the sea rather than governed by higher atmospheric circulation. If I'm right about that then you wouldn't see subtropical deserts down to the water's edge like we have today even if there was land at those latitudes.
Hit me up in comments if there's something else specific you want me to speak to.
Short answer, yes and yes to-a-point. Lets consider Earth, we have a pole centred continent, we call it Antartica, but it only extends to about 60 degrees south at the northern limit of one small peninsula, a third of the way to the equator. To get halfway we need to go to 45 degrees south. Such a continent is going to consist largely of ice fields and tundra but 45 degrees south is just north of New Zealand's Stewart Island so you'll also have some cold temperate biomes and cold deserts caused by either altitude or rain shadow. Biomes that won't exist; anything with "tropical" in the name and the hot subtropical deserts. Anything in high latitude temperate forests and steppe/glassland is possible.
Major note don't live on the coast, such a world is going to have some truly catastrophic storms since cyclonic systems can put in literally thousands of miles of travel over hot tropical waters gaining strength before they're forced north or south by blocking pressure cells near the equator.
Sure you could. Earth was like this once. Here is Earth in the middle Triassic.
At this point the single land mass is called Pangaea and the single ocean the Tethys. What will become the continent of Laurasia straddles the north pole and Gondwanaland the South Pole. You can walk from pole to pole.