I've seen this worked out before, so this post will derive heavily from others work - work that was done for a 32 fighter craft not a 12 fighter craft (which dramatically worse because on 12 airframes you can generate a CAP or a strike, but not both). Cost on that hull was estimated to be about 30 Billion USD (or 5 CVNs, or 30 AVNs) with a top speed of 14 knots submerged, 10 knots surfaced.
I assume the submarine would have to be of significant size to house even a single small fighter-bomber, but how big would one need to be to house a squadron of fighters? (By squadron, I mean roughly 12 aircraft.)
Very large. The least-bad way of doing this would be to raft together a series of pressure hulls about the diameter of those used on the Ohio class boats. I'd go for not less than four hulls wide and one two hulls underneath (likely the lower row would be either two or three hulls wide). You are also looking at a sub substantively longer than an Ohio.
What sort of aircraft today would be ideal for doing so? (If there is one.)
Either STOVL aircraft (Harrier, JSF) or a seaplane (Convair Sea Dart).
I'm guessing that you could launch an aircraft off a submarine with a catapult using the same basic principles as with an aircraft carrier, but I'm no expert on the subject.
Problematic to do. Better to avoid the catapult all together.
Assuming that you did have a submarine large enough to house a squadron of aircraft, and you could use a catapult to launch said aircraft, how would you land the aircraft back on the submarine after a mission?
Either land on water beside the craft or vertically on the craft.
I see. An interesting concept, but hardly a practical one. You mentioned that a spherical shape is the perfect shape for a submerged object. Does the size of said submerged object matter?
Spheres are the strongest shapes, but for storing equipment that we want to remove a cylinder is better.
Size of submerged cylinders is limited by the quality of the steel used. While we can use something like HY-130 on the hull (rather than HY-100 used on the Seawolf and HY-80 used on the Ohio), we're going to limit the dive depth to no greater than 400 feet - so we can close the ends of the tubes to get the planes in and out. This means that a 50'-70' hulls are possible.