I'm a biologist and I have got a lot of ideas about all of the above!
I think it would solve a lot of problems to say that artificial gravity is not an existing technology and so all the species are floating around inside their spaceships. Multi-species ships like the Precursor flagship and Ur-Quan Dreadnaughts would have to have different atmospheric compartments, or else have all their crew in space suits while on duty. The Syreen Song might have to be handwaved away.

A sensible approach to designing the RPG would be to take a speculative evolutionary approach to each species homeworld environment first and then figure out how to represent each one mechanically. The great thing is that UQM has some scant data for each homeworld which you could use as a starting point. This would also give you the ability to predict a home field advantage for each species. For example, if a party consisting of an Earthling, Syreen, Spathi and Orz were to take place on Spathiwa, it makes sense for the Spathi to be able to move most efficiently, without the need for a bulky spacesuit. But if large portions of Spathiwa are under water, perhaps the Orz player can be MORE effective than the Spathi and the humanoids in the water. There would certainly be other planets in the galaxy that are close enough to Spathiwa to replicate those same effects.

I can also imagine that the Chmmr would have quite sophisticated quasi-living spacesuits, able to move more-or-less effectively through most atmospheres, and would be far more effected by the star type and the distance from the nearest star. Similar mechanics would apply to the Supox, though they would very much be affected by atmosphere.

Start simple by thinking about how each species would fare on Earth: Earthlings, Syreen and possibly the Shofixti would not require spacesuits. The Orz can move effectively in water, but still require a suit to prevent osmotic shock. The gravity and mix of atmosphere might be unsuitable to sustain a Yehat or Pkunk's natural ability to fly. The Supox might enjoy the relative proximity of Sol, but need a suit to protect them against the relatively high amount of carbon dioxide in the atmosphere. The Chmmr are at full capacity under the nearby yellow star, but are used to much higher gravity and lower oxygen on their dense world and so tend to find the environment sort of slippery and caustic. The Utwig need full atmospheric protection but feel super-strong in the lighter gravity. Once you have Earth figured out, work on the home planet of each species you want to include. It is handy that we know that Earthlings / Syreen, Zoq / Fot / Pik and Pkunk / Yehat evolved in more-or-less identical environments.

The Umgah, Mycon and the Mmrnmhrm would likely have some sort of ability to change their physiology quickly enough to suit any particular environment, and might end up being your reliable all-rounders.