Drywall – Is it okay to put rigid sheet foam insulation directly over studs before installing sheetrock (fiberglass insulation would be in the walls)

It's common to do the reverse...clad the outside in foam insulation to prevent thermal bridging from that side.

Your idea is sound. I'd suggest using full panels and then tape them. That would add additional insulation to the entire wall and also act as a Vapor barrier for you.

Another technique is instead of using insulation, install 1x2" slats horizontally across the studs every 16" and install the sheetrock to that. The theory is the same in that the only bridging would then be and the small points where the two pieces of lumber meet.

It depends on where you live and how you plan on attaching everything. For interior insulation you want the vapor barrier on the warm side so the colder regions would always do the interior facing and some warmer climates would do the exterior side of the insulation. The underlying rule of where would the moisture most likely be trapped still applies but it typically biased because of installation, how you plan on attaching everything. For interior walls, no vapor barrier is recommended if you use a full fill spray foam because it is a close cell foam (over 2lb density like most spray foam) and is meant to be a tight squeeze unlike most insulation and stops airflow instead of slowing it down. The same can be true for rigid foam especially for outside insulation. If you plan to use contact adhesive, liquid nails, etc glue to hold the rigid board to the house then you'll be essentially stopping all the air flow and that side would not be a candidate for a vapor barrier, additionally if you are gluing the siding on, the same rule would apply.

To sum up: If you are in a cold climate, even seasonally, it is a good idea to put a vapor barrier on the outside of the outdoor insulation as long as you are not gluing the next layer onto the insulation. This will help the trapped moisture wick out better.