Disappearing blue dots

Good call. Yes, I think so. POP is old-style email, when you download the messages to your computer, and read and delete them there. Each POP client works independently of the same account. IMAP is the new method, where mail resides on the server, and clients share a common view of that mail collection. It sounds like the IMAP server is marking the mail as read when your computer downloads it from the POP server.

You may want to change your account settings on the new iPad to use POP, since you're accustomed to it. First, you need to go to your provider's help site and get all the settings, server names, port numbers, SSL required, etc. Actually, you can probably get them from the Mail settings on your iMac. The next step is a little tricky, because Apple tries to make it very simple to configure mail. I'm assuming you're using a big email provider, like gmail or outlook. You need to delete your account from Settings > Accounts & Passwords. Then start to create a new account, but use fred@bogus.com. Apple doesn't know how to set up this account, you say go ahead anyway, then you can select POP. After that, correct the false email address and enter the rest of the settings.

Good luck!