Does Android and Iphone need Antivirus ?

Actually no, almost all viruses and spyware are almost exclusively in the Windows environment.and very rarely get in a Linux based system. The odds to get a virus are slim unless you purposely go looking for one to download.I have had an Iphone and Android system and never have gotten any virus nor do i know anyone who had one in all the years i have been in the computer business.

Should I give out my User Name and Password from my cable account?

Some websites ask you for your account information like your user name and password from another provider.
An example is TNT Turner Network Television website. They ask you for your service provider account user name and password before accessing their website to watch videos.
 You never ever give out this information. With this information they will have access to your cable account and can view all your account settings and change anything on your account. You could get service changes that you never know about.
 They may have a disclaimer about they will not use the information but never trust it.
No legitimate secure website would ever ask for this personal information, so if they ever do move on to another site and warn your friends as well about the site you found.

Do I need super fast internet for paying bills online and checking email?

The answer to this is no. Most people can use a basic package from your local service providers. Check for the lowest packages around speeds like 1.5 megs a second which  is more than enough.
Don't let your local provider talk you into something faster for just checking emails because anymore than this is just a waste of money and they usually get something out of it for selling you a higher paid plan.