I have a wood stove in my basement too but it's not hooked up. The reason? I spent the money on insulation. Did quite a bit of research and, as much as I like cutting and splitting, I don't want to heat the outside. I glued and screwed 2" foam to the interior concrete walls and I'm going to stud a wall down there. I think you can use the foil faced stuff too and I believe with that stuff you don't have to cover it with drywall.
You are wise to insulate. I did the same as you with foam board.
You would definitely need drywall over ANY type foam board. Except Thermax type, I believe.
The gypsum gives the required fire protection (delay).
Foil is mainly for vapor barrier. And if the walls get damp from the outside it is probably better without the foil... then any moisture can dry to the inside as foam board is "semi-permeable" .
You can use foil faced foam board for the rim joist as it is above ground.
Use XPS foam for the concrete walls. (covered with drywall)
http://msdssearch.dow.com/Published...foam/pdfs/noreg/179-07873.pdf&fromPage=GetDoc
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