Who actually provides your Internet service?
Your phone company, your cable TV provider?
Or is AOL calling themselves your Internet service provider when they are actually no more than a 3rd party using the actual trunk lines of the phone company?
If it is either of the two phone company examples or your cable TV provider, they two offer Email services with the Internet service.
But, while that would give you an Email addresss like, for example: YourName@phonecompanyX.net, that does not mean that you have to actually use their webpage Email resources.
You can get the Free Email client application for your PC, Thunderbird (published for free by the Mozilla foundation) (operates much like Microsoft Outlook), and make use of the POP account Email set up between your Internet service provider and Thunderbird; in which new Emails to you get downloaded into Thunderbird on your PC, and when you send Email with Thunderbird it is sent out through your Internet service provider - without you, in either case, personally going to the web-page version of your service providers Email system.
I hope this helps tell you what is possible, when you divorce useless AOL.
But, do save all your contact info in AOL, so you can send notices to friends and family when you have a new Email address.
They don't have to be downloaded. Downloading will occur if you utilize POP protocol, but with IMAP they normally will not be (except as HTML or text for viewing purposes). Several years ago I decided that I really didn't need most messages downloaded anymore and so changed to IMAP. Since I have two big email providers both backing up (so there are two separate backups of each message), why do I need to? But if you really want some messages downloaded you can do it on a folder by folder basis (with Thunderbird). This might make sense on the highest-value messages.