3x8 to 1x8 (with expansion to 1x10/1x11 in the future)
It's unclear if you mean versus other standards, or if there are
different kinds of BSC/ISO (BSA is kind of a misnomer, and in
my opinion it's clearer just to call it all ISO, because that's the
standard that current examples are actually made to). Some road
bikes are still made with Italian, but only a couple brands, and
bottom brackets can still be had for all the old national sizes.
Other than that, yes ISO is the only threaded standard left. Each ISO
BB is made for a certain shell width, or widths in the case of the common 68/73/e-type ones that come with 2.5mm spacers you use as needed, but there aren't shell
interface differences within ISO beyond that.
Yes, no problem.
Chain retention tends to be acceptable with just a narrow-wide ring and nothing else helping it, but it's something I'd be looking at when test riding the new setup. Of all the many rear derailers in the world, you could probably come up with some that didn't pull on the chain adequately. It's also good practice to start a new chainring out with at least a new chain and ideally also a new cassette. (This is now less true of the cassette since suddenly it's much more common for people to have extremely expensive ones.) You could skimp on it if they're still pretty fresh, but the idea is that minimizing wear on the ring tends to save more money ultimately. Worn chains can bring a new chainring to their level of wear fairly rapidly.


Chainline is also something one should look at when putting a 1x or 
any other drivetrain together, although it's perennially overlooked 
and so parts manufacturers do what they can to make it work out okay
passively. Buying a dedicated 1x mountain crank with 49mm chainline 
such as the one you link to and putting it on a normal 135 8/9/10
rear (which wants a chainline somewhere around there) puts you in the camp of not
having to worry about it. Other setups create some hoops to jump
through.

Yes, presuming you get the correct version of the BB. (The BSA 68/73 unless it's got an 83 or 100mm shell.) You need an external BB cup tool.