All right - since we're painting them on, we need to start outlining different types of biomes, and all the variables that they entail.
Biomes (in real life) are determined by physical factors, such as temperature range, precipitaition/liquid availability, as well as ecological factors, which determine what lives there.

Physical Factors
Temperature: we need it in there somewhere, to make sure that physical properties remain consistent, but it will be largely under the hood.
Ground vs. Groundwater vs. soil type: As far as I know, we're flagging different groundcover types with values which will determine whether they are sand, rock, soil... presumably, these tags would include a variable about water retention - whether they would dry immediately or become muddy and swampy.
Precipitiation: we need to remember to include something for how often and how much liquid will fall from the sky. Some places will rain/snow/liquid methane a lot. Some will not.
Elevation: Except in specific cases, this consideration will be modified by temperature. Still, there are some biomes that will need flat land, to be underwater, or to be just above the water table, such as a cave.

Ecological Factors:
This is all about the autotrophs. In real life, fast growing autotrophs inhabit places that are frequently disturbed, whether by fire, avalanche, tides, or whatever, while the more stable the area, the more diverse the number of autotrophs and the longer it takes them to grow or replace themselves. This is why cutting down rainforests sucks for the whole surrounding area.
Some Biomes will be more fragile than others.
How I think we can/should handle this is that certain biomes, which are less-disturbed, will have a greater amount/variety of autotrophs populating them when we just paint them in. This means that competition and specialization in the area will increase: There will be more available living resources, but stronger competition for them.

There's probably bits and pieces of this scattered everywhere across the PPG forum, (and maybe we've already dealt with bits and pieces of this over in programming,) but I did want to explain the variables that would make a biome "realistic" instead of just cosmetic, and open up the thread for further questions/comments/explanations.

We're also going to want variables for underwater environments, such as currents, geothermal activity, etc, as well as factors for elevation.

The factors that determine a water environment could be;
1. Depth
2. Ground (if any) make up. Like, rock seaweed, sand, etc.
3. Water currents
4. Light received (which could probably also go with depth)
There are probably more, theses are just the ones I could come up with off the top of my head.

Oh, and I have another biome;
An underwater environment centered around one massive organism. It would have spongy skin so that some plants and coral-like animals could grow on it. It's basically a giant, moving coral reef. The main creature would stay very close to the surface (about 20ft down) which would allow aerial predators to feed on fish. The creature would go with the currents, so the others would, too.
Would require-
-A massive animal, the max size one can be
-Aerial predators, which should be able to dive, and digest meat
-Plants that can survive under water
-Coral-like animals
-Small plant eating fish
-Medium plant eating fish
-Small predators
-Medium predators
-Large predators that come and go

Therefore, we will have to restrict some biomes by the rendered elevation of the planet and some by whether they are underground, underwater or both.

Unless we have a cave/cavern, we don't have a seperate biome. Soil is the basis of what sits on top of it. Period.

How do we mathematically define where these said biomes are?

For instance:

We know treeA goes in biomeA.

Where do we put treeA?

I think we have a procedure lurking somewhere to randomly place large autotrophs, based on elevations and distance from the others.

Reasonable. Elevation fluctuations may have to play a role too, but other than that, should be good.

Pines like to grow on steep slopes, because they can get much more light in that way. Elevation plays a big role.