Walls – use fewer than the recommended number of fasteners for exterior wall insulation

Yes. There are several types of rigid foam board. Polyisocyanurate (polyiso) has the best R-value at 6.5 (after aging). A 2 inch thick board will be at least R-12.9.

There are two main methods you can use from inside the room. With the first method, sometimes called "Cut-and-Cobble", you cut the panels to fit inside the joists and then use spray foam (e.g. Great Stuff) around each panel to create a tight air seal. If you have a 2x4 exterior wall, you could even get 3 inches of foam in there if you can find 3" panels or use a 2" and 1" panel together. I recently did exactly this myself in a bedroom that I gutted down to the studs.

Another method is to add the foam panels to the inside of the wall on top of the studs, tape the panels together with foil tape, and then put the drywall over the panels. This second method is preferred since it reduces thermal bridging from the cold studs into the wall and also provides a continuous air seal across the entire wall. If you do put the panels over the studs, you must fasten them to the studs and then use longer drywall fasteners to connect to the studs since the foam is not considered structural. The downsides to this methods are that all of your trim and receptacles must account for the additional wall thickness and you lose a bit of space in the room, depending on foam thickness.

If you wish to preserve the brick exterior, you have the following options, depending on the wall construction:

1. If you have a structural brick wall (e.g. multiple-wythe brick or brick veneer on block) then you will have to insulate the wall from the inside. This will entail applying rigid insulation board--either foam or mineral wool--over the inside of the wall, and then covering that up with drywall.

2. If you have a brick cavity wall (e.g. brick veneer on the outside and structural brick or block on the inside) then you can blow water-permeable insulation like perlite into the cavity. You could additionally insulate this wall from the inside as per above.

3. If you have a brick veneer on wood framed wall, then you'll need to insulate the stud cavities normally. You can use fiberglass batts, mineral wool batts, closed-cell spray foam, or dense-packed cellulose. All will be effective. One advantage to dense-packed cellulose is that you don't actually need to take down the drywall; you just drill holes in the top and blow in the cellulose. This could save you a ton of money if you haven't already removed the drywall. If you have to or plan to remove and replace the drywall anyway, my personal recommendation would be installing either mineral wool batts or dense-packed cellulose in the stud bays, covering the open stud bays with an additional layer of rigid mineral wool board, and then applying the drywall over that. An added bonus is that this will all but completely soundproof the house.