yes to almost all your questions...
but...
analog isn't all that easy.
you have to find the resistance range on the pot on the existing joystick... It's probably 0-100k
true arcade analog joysticks are 0-5k. There isn't a way to directly convert that.
BUT... you can do a few things.
This assumes you are going to use a REAL arcade analog controller
1) for about 25 bucks (you need 2) you can get 100k pots to put on the joysticks. So for $50 or so you have an old joystick with new pots.
2) You can give up the controller for analog and get Dave's AKI's controller. Its 5 analog axis's and 14 buttons... And your done.
http://dave.bit2000.com/ and its 32 bucks..
If you plan on using your original... I don't know off hand on how to mount it... but taking it apart + a rounter on your control panel should hopefully do it for you.