Congratulations on all these huge milestones in your life! Wow, that's a lot at once!
No, it won't look too small. Don't worry. I'd actually recommend the 2mm over the 1.5mm. In my experience, rings less than 1.7mm wide are often susceptible to bending, warping, or cracking, and they're just not durable enough (which is a shame, since they can look so nice!).
Well, at the end of the day, more is more, so if size is a priority, then sure, you should do that. On the other hand, price relates to size EXPONENTIALLY, so the more carat weight you get, the more painful the numbers will be. The size difference between 0.90 carats and 1.20 is significant. Between 0.90 and 1, maybe not so much.
Never mind the formal grading for a minute - think about it in practical terms. Look for a stone that's white enough it won't look yellow or dingy, that's clean enough that no inclusions bother your naked eye (it doesn't have to be microscopically clean!), and that's well cut so it sparkles like it should. What grades does that mean? It varies for each individual! Look at a bunch of stones and see what YOUR tolerances are!
Not necessarily... But you need to understand what you're buying. IGI's grading standard (and AGS's, and EGL's) is different from GIA's. Most experts agree that GIA has the most strict grading, so you wouldn't expect an IGI H color (for example) to be as white as a GIA H color. But you also wouldn't expect to PAY the same for it. So, if you're buying an IGI stone, make sure you're paying the IGI price - don't pay GIA price for IGI stones (or AGS, EGL, etc.). Does that make sense?
Look for a stone with good proportions (square, not rectangular, not too deep or shallow), and avoid a girdle that's extremely thick (makes the stone look smaller and reduces light performance) or thin (makes the corners very vulnerable to chipping). You can google the "ideal" specs for a princess cut all day long and hear a hundred different "experts" insisting their formula, or their "super ideal" cut is best, but at the end of the day, you need to look at some actual stones (in real life if possible, via an expert proxy if not) and see what looks best TO YOU.
Again, congratulations, and good luck!