1. You will need to add a payment method before creating your first droplet. You will need either credit card or PayPal info.
2. You are purchasing a virtual server, so you can host as many domains as you want. You just need to either enter them in the DNS panel or use something like any DNS service provided by your domain registrar or a 3rd-party service like DNSExit.com or CloudFlare to add A, MX, etc. records for your domains.

Thanks a lot.
One more question:
Do I get one dedicated IP or do I get a couple of IP's?

Each droplet comes with only 1 dedicated IPv4 IP address.

Q: What form of payments do you accept?
A: We accept Visa, MasterCard, American Express, Discover, and PayPal.

I'm thinking of getting the $5/month plan to start. Does that also allow as many domains as I want? Are all the plans that way?

@joseph.yuhas: Yes, you can host as many domains as you want on a single droplet.