As I said, to find/prove the generic intitial ideal you have to do a generic coordinate transform and compute the initial ideal. Whether this can be carried out depends on the ideal that you are giving. If you want it for just one ideal, then you can choose a random coordinate transform and with probability one you are fine. If you want a class of examples, like "generated by two forms of degree $d,e$ then you need to do it in general. Note that the code that I gave is that 'more general'. It proves the statement in the book, it is not computing an example.

Maybe the connection equivalence of unimodular triangulations and square-free initial ideals is already the best argument why triangulations are 'better'.

I guess I don't see what you question is. The smith normal form of a given matrix exists and can be computed using elimination. If you are asking for the complexity, then that is measured usually in ring operations, so it depends on how fast you can do ring operations. I don't understand the meaning of the sentence "The presence of a grading seems to imply one should take some minute care".

You have used $U$ twice... but well.

Excercise 3.5 in Eisenbud's commutative algebra book may be related. It asks, among other things, to show that all associated primes are homogeneous as soon as the monoid is totally ordered abelian and shows a counterexample if not. This feature is independent of the characteristic. However, in the radical the non-homogeneous minimal primes can 'fit together' again.

Thanks! In most cases being a $k[S]$-module direct summand is what one is interested in. I originally though of a direct summand of rings, is that a meaningful question at all?