I actually disagree with @Twelfth's answer to this question. The biggest regulators on planetary temperature isn't the high heat capacity or heat of vaporization for water, but rather the circulation of the atmosphere and ocean. By moving hot material away from the equator and toward the poles, you keep the planet a much more consistent temperature.
To make such a world possible with that consideration, there's two things that move heat around on planets like this- atmospheric circulation and oceanic circulation.
Solving the atmospheric circulation problem is easy. All you'll have to do is make it a bit smaller than Earth and slow down its rotation. This ensures that there's only two (one north, one south) large circulation cells on the planet and thus an efficient transfer of heat from equator to poles.
This would actually be aided by the concentration of landmass at the equator- land heats up much more quickly than water, so the rising branch of our circulation cell is strengthened. You've got the smaller planet part covered, it sounds like, because if I understand you correctly then your planet has a circumference of 2,000km instead of Earth's 40,000km. The slower rotation speed reduces the effect of the Coriolis force, which is responsible for breaking up the circulation cells.
The oceanic circulation is a bit more complex. The thermohaline circulation on Earth is powered by deep water formation in seas near the poles, as the result of cold, salty water being very dense and sinking. This pushes water all through our oceans and transfers heat effectively away from the equator. On this theoretical planet, it's hard to make brine via ice exclusion, but we could still have deep water forming as it cools, and sinking at the poles- another circulation cell to regulate planetary temperature. This would be impacted strongly by mid-ocean ridges, but as Twelfth points out, there are going to be bigger problems with continental plates on this planet. I think the oceanic circulation would be a lot weaker than it is on Earth, but would still be moving a significant amount of heat from equator to pole. There would be weak, constant upwelling along the coasts of the landmass, bringing lots of nutrients to the surface and fueling a really productive ecosystem in the ocean and the sea, and a weak world-circling current that wouldn't be detectable to someone in it.
As for the actual question, relating to flora and fauna, I'm not sure they'd be all that different. You might lose some diversity because there are no separations between continents (goodbye, kangaroos) and because you'd only have a rainforest biome. As you point out, they'd probably be better adapted to high temperatures, and if you slow down the planet's rotation then they'll have to be good at regulating their temperatures as they are rotated into and out of the sun. Plants would require larger energy stores to deal with the long periods out of the sun, but animals would still be able to eat like normal. Ocean creatures would be very much the same, although I suppose it's possible that you'd have less plankton and more nekton because phytoplankton would want to avoid being dragged down at the poles and wouldn't be able to hang out in gyres like they do on Earth.
The above commentary assumes that life began on such a world, given that life on Earth probably started in shallow seas or at hydrothermal vents, but that's a different question entirely.
tl;dr: Yes, such a planet could absolutely exist and isn't actually that hard to imagine. Flora and fauna wouldn't be that different, maybe with a drop in biodiversity due to lack of separation and a single biome.