This is not a direct answer to either of your questions, but I don’t have enough reputation to comment. Perhaps the recent preprint of Booms, Erman and Yang might be of interest to you: https://arxiv.org/pdf/2007.13914.pdf
The title references Veronese syzygies (Edit: looks like the title changed in a recent update and no longer references Veronese syzygies), but the heuristic developed is based on syzygies of random Stanley-Reisner ideals.