Minimal Deterministic Buchi Automata Product
This is only a partial answer (as I believe that the state of the art is insufficient to answer your questions completely - I am happy to be proven wrong here), but I hope that it helps you anyway.


First of all, note that when taking the conjunction of two automata with $n$ and $m$ states, the result having $nm - o(n) - o(m)$ states cannot be avoided. This follows pretty much from the fact that for deterministic automata over finite words, such a blow-up can also not be avoided. At the same time, we have an upper bound on $2nm$ many states for the product. For automata theory researchers, this is actually pretty tight.
The constructions given in the Vardi paper and the one given in the slide set you referred to are slightly different. Both have counters for keeping track of for which automaton an accepting state has to be visited next. In the slide set construction, after both automata have seen accepting states, goes to the counter value of 3, where in the product an accepting state is visited and then switches back to 1. Vardi (who attributes the construction to earlier work by another researcher) does it differently. In a nutshell, the idea is to recycle the accepting states from the factor automata, and to switch the counter only after an accepting state has just been seen. In this way, the "3" counter value is not neccessary, and the construction still works anyway.
As far as I know, there is no research that answers your questions completely. But you may be interested in Schewe's paper that shows that minimizing deterministic Buchi automata is NP-hard (The ArXiV version of the paper can be found here). I would not be surprised if you could base a proof that your problem 3 is NP-hard on his proof ideas. For the case that the factor automata are not already minimal, this proof would be trivial, but I assume that you want to start with minimally-sized automata.