welcome to the madness!
definitely take your time and learn the differences between the case shapes, the models, etc.
as far as your domed crystal question, no, all models do not come with one. one of the best modern references, the 233, does have it.
if you feel like getting a headache

, other questions to ask yourself include:
* radiomir, luminor or 1950s luminor case?
* what size? best advice here: go and try on lots of models....some 47s wear more comfortably than 44s, depending on wrist size and shape.
* manual or automatic?
* in-house a deal-breaker for you? or, are you okay with top-of-the-line and well-tested ETAs?
* what material for the case (SS, Ti, gold, ceramic, composite, etc.)?
* basic functions/dial, or complications/busier dial? in the panerai world, even a seconds hand is considered by some to be too busy (not me, though).
* is a date necessary?
* are you okay with a cyclops on a PAM?
* sandwich (lume markers recessed in holes cut in dial) or sausage dial (painted)?
* what color dial? (black, brown, white)
* what color lume in the daylight? (green, ecru)
* what color hands? (black, silver, gold)
* strap or bracelet? for a PAM, really, straps are the best way (and very addicting), but, still a question to ask yourself.
* is the quick strap change system important to you? it's pretty much only available in the luminor contemporary line.
* are you okay with plexi (especially relevant to new models like the 372)?
* do you want a limited or special edition?
* what about vintage?