So, keeping the IE/OE e-mail address would make life a lot easier and who couldn't use that?
Again to clarify: your email address isn't associated with Interner Explorer, Outlook Express, or your old computer. It is issued by the cable company and it doesn't change if you get a new computer. You will be able to check the same email address using software on the new computer. Your new computer will have the OS X programs Mail and Safari (browser). You'll configure Mail to pick up messages from your existing email address.
I like to use this analogy: Having an email address is like having a post office box. Your email messages do not come directly to your computer; they are loaded into your post office box (the post office box is located on the mail servers of your internet provider) and wait there until they are collected in some way. Collection can take place with a variety of programs--Outlook Express, OS X Mail, etc. These programs remove the messages from the post office box and load them onto your own computer for further handling.
I don't know much about AOL, as I haven't really used it since version 1.0 or so. I would think there is an OS X version of AOL but I don't know for sure.
As for the Excite account, this is a 'webmail' account, isn't it? That is, you check your messages by visiting a web page and logging in, right? You should be able to do this from any computer as well--that's what webmail is designed to do. Not sure what particular feature wouldn't be supported on certain computers. . . can you explain a bit more? I'm still not exactly sure what you're referring to when you talk about 'email notifiers.' You're meaning a program to collect mail, right? As stated, you will certainly be able to do this on your new computer.