The Internet in general will work just fine from 10.3.9. You may see some odd things in Safari since it's so old, but tha vast majority os stuff will work just fine. My first PowerBook had 10.3 and it ran fine with my Comcast connection along side my Linux and WinXP systems.
As for files, I assume you mean emails? If so, I also assume you are not using Mac Mail? If not, I would recommend you start. Add an account for your old ISP account (assuming it's still available) and download the email you have stored on their system and sort the messages into whatever folders you deem necessary. Then add an account for your Comcast email address and start getting your messages there. Then deactivate the old ISP account (just stop Mac Mail from checking that account, don't delete it if you have anything in that inbox).
Mac Mail is a great email client with a robust set of features. It may look and act a little different than what you're used to, but that's to be expected and you will get over that very quickly.
Yes. (and no). At random intervals, they "improve" their software by adding features. Some of these may not work very well if you want to read Comcast pages. If you stick to going to other sites, there should be no significant problems. Do not install any Comcast software on your computer!
As far as transferring email, I would suggest saving it on your computer using Mac mail. Then if you want to also save it on the Comcast server, just email it to your new Comcast address.