I am removing the siding on my house in Montana, which was built in 1964. From inside out I have sheetrock, 4" studs with R-11 paper backed fiberglass insulation and gypsum board sheathing. I want to add 1 inch rigid foam insulation and reside with steel siding. Should I install a vapor barrier between the existing sheathing and the new rigid foam insulation?
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.
Yes. I would use a form of home wrap. No brainer in Montana.
And I usually use Tyvek(?). Not sure of other have opinions on the best brand or style.