Ok, POP3 is the account type I'm least familiar with; so bear with me.
When you are looking at your account settings, there is probably an arrow to the right of the email address in the Account field. This will take you into more settings; including the Advanced settings mentioned earlier. You won't find the days to sync option. That is exclusive to Exchange accounts.
The nature of a POP account is to download emails onto the local machine, and deal with them there. For that reason there should be an option to delete emails from the server on download, or leave them on the server. This would affect whether other email clients would see a new email. It should not cause emails to disappear from the iPad; however, if you use another email client (on a computer or other mobile device) try changing it's settings to leave the emails on the server, and see if that helps.
At least one client needs to delete emails from the server, or if you have web mail access that would work. POP servers usually have a limited amount of space to store email, so it has to be cleared out eventually. If a poorly configured POP3 server were to get too full it 'might' delete email in a randomish fashion, instead of starting with the oldest emails first. As I said, I just don't' know enough about this type of server.
You're provier may offer more than one type of email server. If it does and you are mixing IMAP, Exchange, and POP clients, things could get unpredictable. So check through all your devices and computers for inconsistencies.
That mishmash of advice is about all I can offer. Hopefully someone who is more familiar with POP3 can offer something more helpful.
And I never asked how much email you are keeping in your Inbox. Keep in mind that there is a setting (Show) that limits the number of emails that will show in your Inbox. You'll find it lower down on the Settings > Mail, Contacts, Calendars page. When you are using IMAP, you can always choose Load More at the bottom of the mail list if you go over the Show limit. I don't know what POP does.