Can the registrar remove my domain whenever they wish or are their any conditions?
It shouldn't. But don't choose GoDaddy
There are no requeriments. Just go to any decent domain seller and get it. There are some which let you pay with paypal, librety reserve, or even bitcoins. That could be an advantage if it is for extrange things, you know.
Then you will need some hosting. Most of them have their own DNS so you can configure your domain to just use it. If not (that was my case), you can
a) Install a DNS server in the host and make it resolve to itself
b) Use some free DNS service such as dnsExit.com