Hi Jen - I hope it works for you. Just a note - I don't know that you have to mark all the emails on your server as "read" - if there are ones that you want to keep as Unread by all means do that.
I believe (just guessing from my experiences!) that whatever is in the Inbox is what downloads to a new mail account set up on an iPhone. Unless of course you are setting it up as IMAP instead of POP. I believe IMAP makes the server and all other devices mimic each other. So I assume if you made a folder on your server called "Money" and moved emails into it, that folder would also appear on your device (and visa versa). I DO NOT use IMAP so I can't swear by this, but managing my boss' wife's email on her iPad (which is a gmail account) makes me believe so.
If you IMAP, even if you "read" the mail, it will download on your phone to mimic the set up on the IMAPed server. So if you have an inbox with 1,000 emails in it on the server, that will try and come onto your phone. If they appear "read" on your server, they will on your device.
I prefer POP because I use my phone to see if I got answers, or have a check on email at all times, but I can delete on the phone and then its still waiting for me when I get home or work.
Hope I'm making sense. I have no idea why Apple took away the ability a couple of years ago to choose "only show 50 emails" on the phone as it was quite handy.
My suspicion is that the vast majority of users are younger generation who don't care about keeping every documentary email for the past decade. For someone like me with a legal and admin background, I keep everything! And for the last 30 years I've been used to using my Inbox as a "to do" list. When I'm done I delete or file it to a mail folder for keeping. So I never have an Inbox with 500,000 emails in it, but know many people who do...
Hope this helps you!!!! Good luck 🙂