Something to keep in mind is your email service is a classic 'server - client' arrangement. Be it Gmail, MS Outlook, Yahoo Mail, etc. they all have that functional similarity. In your case, your actual Gmail account (email, calendar, contacts) reside in one of Google's online servers, and the Gmail app on your phone is simply the 'client' that gets all its data from your online account. Google stores your online data, and does all the management and maintenance on its end. Your Gmail app is just how you interact with your mail/contacts/calendar. The app does manage your Gmail account data on your phone, but it's still completely dependent on whatever it does have is from your online account -- i.e. you can use the Gmail app to manage your emails on your phone when it does or does not have an online connection, but of course once you need to retrieve or send an email your phone needs to be online. Once you understand the relationship between an online email service and a smartphone email app, that makes it easier to deduce mail issues and such.
I don't use the Gmail app but have you checked the app's Settings menu for any options regarding how email messages it stores? Two weeks sounds like a typical default that might work out OK for some but does seem to be a short time period. If there's no such option in the Gmail app Settings menu, you might want to think about switching to a third-party email app instead. A non-Google app will tend to have more extensive options to customize and optimize your email usage to match up with your own needs, not just what Google wants you to see.
My own personal preference is the K-9 Mail app:
K-9 Mail is a 100% free and open source email client for Android.
play.google.com
It has a very extensive Settings menu with a long, long number of options, that in your case include a number of options for which folders to view and how many messages to retain. (... note there's difference between time period, two weeks, and number messages). So can set it save as few as 10 emails or as many as 10,000, or all.
K-9 Settings also have a lot user interface options too. A drawback being its Settings menu might be a bit too complex for a casual user.
Some options that might be more sutable
The Aqua Mail app gets a lot a good user reviews:
Email client for Gmail, Yahoo, Outlook, Hotmail, GMX, AOL, IMAP, POP3, Office365
play.google.com
The Blue email app is very popular too:
Mail support for all email accounts: IMAP, ActiveSync, EWS, POP3
play.google.com
Anyway, don't rely upon the defaults in whatever app you're using, take the time to dig into an app's Settings to set up up accordingly.
Regarding POP3, are sure your Gmail app is set up to default to POP3? Confirming that is cumbersome through an app, it works out better if you log into your Gmail account via a web browser using a desktop or laptop to check its base settings. Go here:
Log into your account and open up the Gear icon in the upper right, click on the 'See all settings' link, and in the resulting page click on the 'Forwarding and POP/IMAP' tab. That should show support for both POP and IMAP are enabled by default, along with various options as to number of retain messages and other related items.
Note that is a way to access your Gmail settings from a more foundational aspect, there's likely to be a way. I just haven't encountered a smartphone email app that allows the same level of access.
But whatever the case, don't dig yourself into a dark hole of mystery. If you want to have more messages saved than just fix the problem. Don't rely upon Google's Gmail app for anything more than basic usage (and confirm that your Gmail service is indeed using IMAP).