Do you accept my answer?

I have clarified the text.

@M.Dus. As it is a sum of two lengths, $f_1$ is positive.

In positive characteristic the only connected groups of finite homological dimension are the tori. That is because we know enough about cohomology of the additive group or about the cohomology of Frobenius kernels. Then induce up in the sense of algebraic group representations.

It seems you are asking if $K_2(\Phi,R)$ is always generated by Steinberg symbols. Definitely not.

@DavidG.Stork Questions 2 and 3 would take more. But notice that my code accepts any initial data. So one can experiment to get a feeling. The code will be cleaned a bit, by the way.