Doors – Which way should a door between the garage and basement swing

If the existing setup works, I don't see any reason why it wouldn't work with new doors instead. One down side to this setup though, is that doors without jambs aren't that secure and could probably be kicked in fairly easily. If it were my garage, and I didn't want a standard fold up door or a roll-up, I'd probably put in 2 larger barn style doors, or if you have the space, one of those sliding barn style doors that has an overhead track to slide on.

Your idea about the 1X6 on either side is a good start. It would give you something to fasten to, in a simple fashion on both sides, although the fastening would be minimal. You may need to increase the number of nails in the jamb to keep it in place over the long haul. Same thing if you use screws.

When the 1x6, for a 6" wall, or if it fits, 1X8 for an 8" wall, is set in place, use construction adhesive between the block and 1X and let it harden. Make sure your block are clean so the glue sticks. This will give a firm base to work to. Do not skimp on the glue. Use a 10 oz tube, a half tube at least, on each side before you set the 1X. Use concrete screws to fasten it in place, they are more reliable for a first time installer. You could use masonry nails but there are places they may not hold. The screws may fail in their holding, but that will be obvious. To increase the chances of holding, use the screws only in the block, not the joints. Keep the screws about an 1 1/4" to 1 1/2 from the edges of the block. The block is thicker there and is far enough from the edge to keep from cracking out (should be). Screws also only affect a small area when they fail, nails break out larger pieces.

To answer the other part of your question, yes you could use framing only on one side, them shimming and fastening the other side is tricky. There are case hardened finish nails to do this, or even screws. For a first timer, what I mention above should do what you need.