Its looking good.
Generally you can fill gaps with expanding foam, low expansion foam, bits of foam or insulation, or just more plaster and scrim tape to stop it cracking
What is "foam applying"?
Expanding foam is a gap filler more than anything, not for insulating surfaces.
You may need a quick setting and low expansion product instead
You'll have to wait and see. It's the wrong foam for cavity fill
Cavity foam injection is a different foam, with different properties, expansion and cell formation (not hollow like this canned stuff)
You should have just pointed the gap up with mortar or a mastic
Yes fill any gaps with expanding foam for larger gaps, or silicone seal or caulk. Or whatever really. It won't matter to the wall or floor.
Don't get hung up on wall moisture balancing.
Foam can push the insulation board out and created gaps or uneven filling. Its OK for joints.
I can't really understand what the problem is, as a board or boards of insulation can be cut to fit between those timbers and foam is not even necessary.
Ram some mortar in it really tight and full. Or likewise with slate.
Not foam or anything that can compress or shrink. If the gap is full the bricks wont move
Push some mortar, knock some wedges, or squirt some foam into the gaps if they are more than 10mm, to stop the board twisting. No need for extras fixings
That's a small gap.
For the cost and time of foam, mastic/adhesive and trim, you may as well just squirt two tubes of mastic in and smooth it with your finger.