I'm currently running a RH9 box at home that serves as my web and mail servers along with some other things. It's not going to be easy to sum up how to accomplish everything you want to do with a few paragraphs, so I'll do the best I can while making it as short as I can.
In reponse to your first question, (a) webmin is an excellent app for managing your linux box. However, by default, by logging in remotely, you're transmitting your password without security. Once you get webmin installed, be sure to read the help files on how you can secure it. (b) You don't necessarily need webmin in order to reboot your linux box. You can also use ssh which is a lot more secure. And if you don't have apache installed, use the red hat sofware manager and install it. Check the apache documentation for the correct configuration.
When you say, "key in the login information and click to connect to the internet", are you implying that you're on dialup? If you're on a broadband connection, you shouldn't have to manually "connect" each time you reboot your box. Once your network settings are configured for your box, it will connect automatically whether you're logged in to your computer or not.
Suggestion: If you are on a broadband connection, I would suggest putting a firewall in front of your linux box, whether it's the RH firewall or a broadband router. You can NAT your public IP to your linux box on specific ports so that everything isn't completely open to the outside.
At minimum, you'll need:
port 80 - for web
port 10000 - for webmin
for what you're trying to accomplish.
As for your integrated video/nic card, are you sure you installed the driver after you untar-ed it? And even if you did, you'll probably still need to edit the config file located in /etc/pcmcia and the another one located in /boot/module-info-*kernel#* (probably module-info-2.4.20-8 for RH9) in order for it to show up.
I know that all of this is pretty generalized since there is a lot more to it, but hopefully this can get you started.
My personal suggestion is, if you haven't yet installed apt, go to rpmseek.com and download apt for your system. Once installed, fire up your console and run "apt-get update" then "apt-get upgrade". This is will update your system, and that's where you should start first.
Good luck.