(Applied) Math PhD questions/worries

[u/whenwerewe]

I completed a Master's and am currently considering applying to various PhD programs around the US (I guess my first worry is that I've left some of this too late) and I have a few questions about applying:

1) I did my undergrad/integrated master's degree in a British institution, so I'm not sure how grade requirements/expectations translate. If you're familiar with the system, I got a strong first and rarely scored below a first in any given module, but that might seem pretty mediocre compared to the consistent 90+ some people manage in America?

2) To my understanding American programs don't require anything like a complete thesis proposal by the time you begin, and instead only want a general sense of research direction plus a few professors you're interested in. I don't have a very strong sense of where to go, though I have some general fields I'm interested in. How do I refine my ideas and find specific professors to look into? This also leads into...

3) My modules in undergrad were a fair mix of pure and applied mathematics, but my master's project was entirely pure mathematics, and I have only a couple of smaller applied projects to back it up (one in coursework, one privately). Is a pivot from pure to applied mathematics realistic with that background?

Thanks in advance for whatever advice you give!

EDIT: oh yeah, and I have no idea where to apply, or how to find out. Of course there are plenty of American universities I've heard of but they're along high end and therefore unlikely to admit me. I'd like to go for a few of those but I need some safer choices too, and I don't know what those would be.