Hello VBBN!
1) You need to have your modem/wireless network adapter connected any time you want to play online, since that's how your Xbox accesses the net.
2) You can use a credit card to buy Live either straight through your Xbox (on the dashboard, go to My Xbox, select your gamertag and choose to upgrade to XBox Live) or you can buy a prepaid card from a retailer on the net. The second option is usually cheaper, but I don't know if any sell one month cards. Jasper1675 is a good one if you ever want to take the plunge and get 13 months, and they'll email you the code, so you don't have to wait for the physical card to arrive in the post.
3) My pleasure ;)