Don't worry too much about adding random mountain ranges wherever you need them. As long as they are older and lower- i.e. more eroded- then it's easy to explain. Look at the Appalachians or the Anti-Atlas range. Neither is well explained by current plate collisions. That's because in fact they are both remnants of the Central Pangaean Range from almost half a billion years ago. So if you add any random mountains just make them more Appalachian-like and explain them away as remnants of an ancient range. I think that the Urals are also an ancient mountain range remnant.
That said, I think that there is a problem with your plate tectonics. If that is an equirectangular projection, then the plates don't line up right at any map edges. (I understand from my limited trolling around this forum that this is a common tendency among our peers.) If you have a plate cross an east or west map edge then it must continue on the other map edge- just like if you made a continent that did so. And yours don't.
Heck, at your south pole you've got one plate bountary reaching the pole- you can't have just one. In fact, it must be zero or an even number, or it would be one heck of a coincidence that there is a three-way plate nexus precisely at your north pole.
Put your tectonic map into G.Projector and look at it in ortho view to see what I mean. I would imagine that it's easiest (if somewhat contrived) to make both poles their own plate.
Or is this not meant to be a projection of the whole world?
Regarding climate/poles/etc.: is this a fantasy map? SciFi? What? Because you can explain away almost anything with "it's magic." And polar sun mirrors help, too.
If neither of these apply, well, yes even then you can still have a world without icecaps. But there is a price to be paid. The Earth has gone through many periods where it had no permanent icepack- see this Wiki page. In fact, current theory holds that the Earth only hosts a permanent icepack 20% of the time. The Earth goes through periodic greenhouse and icehouse phases on about a million-year timescale- this is different than glacial and interglacial periods, which only happen during icehouse phases (we are currently in an interglacial icehouse). Of course, the average surface temperature is about 10-15C higher during a greenhouse phase- that's the tradeoff. But in any realistic scheme if you want equatorial climates no hotter than modern Earth on an Earth-sized planet with Earth-like axial tilt and rotation and Earth-like weather patterns, then the poles will be cold and probably have an icecap. As we are working on proving it needn't have icecaps year-round, especially if a pole doesn't have land under it, but at least in winter.
But if you are willing to let the tropics fry then you can lack icecaps totally.
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