I think a simple google search for NRA pistol safety course in connecticut will have some results. Or check the NRA website. I found my instructor from a flyer at a local bodega in Hamden. He was sort of a low-budget but legit option. I think I paid $120 and got my certificate 3 hours later. There are more thorough instructors out there with higher fees.
I don't think checks or cash are allowed. Money orders or bank check is fine. Go to your bank and ask for bank checks. Or go to a supermarket and get a money order.
Yeah, it's a few months at least. Mine processed quickly but others may take longer. There are surges after a mass shooting so there might be a longer backlog now. I got mine before Newtown happened so it wasn't bad.
As far as your choice of gun goes it sounds like you need more experience and your NRA instructor should help. To answer a few questions:
You should keep your guns safely locked in your house most of the time but there might be times you choose to bring a gun with you in which case CCW becomes a factor. As far as carrying a gun the car goes, I think you either need to be in the car or it needs to at least be locked up really well and out of side. I wouldn't recommend just leaving it under the seat or glove box. I don't leave my gun in the car unless I'm just running into a place where I can't bring it like a government building.
A revolver is generally reliable and easy to use but it has some pretty big limitations too. A standard 6-shot revolver actually only holds 5 shots because you will will keep an empty chamber for safety. So it's limited in the amount of ammo it can carry. A Taurus Judge is over-kill. If you use it as a home defense weapon it will probably shoot through the intruder, through the walls, through the neighbors walls, through the neighbors, and in the next house. You always have to factor the penetration of the bullet and what is behind your target. There is no great reason for over-kill. 9MM to .45 is more than enough.
Let your wife get her own gun if she wants one. Don't share. What is a good fit for you may not be a good fit for her. One-size fits all is not the way to go with guns.
"It's a classic" is a bad reason to buy a gun unless you are collector. You are not a collector...yet. Your first gun/guns should be functional. That's the whole point of getting your permit: for protection first. Once you are covered there then get whatever you fancy.
AR-15's are more or less illegal in CT now unless you are grandfathered-in or it's a really expensive pre-ban. Don't bother. AR-15's are also not "cool". That is kind of a stupid and dangerous comment to make. It's a weapon. Weapons do not make you cool. An AR-15 or similar weapon is not a good home or personal defense weapon. It's strictly useful for range shooting or military defense. Get the whole "i'm cool because I have a gun" idea out of your head now. The reality is that if you get your permit and get a gun the first thing you should do is be quiet about it. Don't tell anyone you don't need to tell. Don't brag. People steal guns and people also have strong opinions about them. If you value your friendships and want to be a responsible gun owner then act like one. If you're looking for something you can brag to your friends about go buy a Corvette.