Outlook express requires a connection to retrieve or send files. It will complain if your not connected, but you can ignore that and view the mail you've already received.
Another thing is we have some setting we have to do to tell it to keep a copy on the server.
Not sure if this is a question... remember that all your e-mail is not recieved by your machine at all, but rather a mail server. I your case, probably your ISP, for hotmail, these servers are owned by MS. in order to retrieve the e-mails, outlook (or another E-mail program) needs to communicate with that server; this is what POP3, SMTP, etc are for; communicating with this server. That checkbox determines wether outlook will delete the e-mail from mail server after it acquires a local copy. The e-mail will still appear in your Inbox in Outlook, but it's not in your inbox on the server, so it isn't taking up any space there.
I don't believe you can use outlook express to browse the web. it might render HTML in messages, but it will use Internet Explorer for that. When your ISP asks what client your using, that depends on the context: if you phone regarding issues browsing, it's your browser. if e-mail, your e-mail client. And if it's neither you should ask for clarification
Programs like Outlook Express are called POP email clients.
No, they are just called E-mail clients. POP3 just happens to be one (of MANY) protocols they can use to send retrieve mail. There are also SMTP, IMAP, SMAP, etc.