That is indeed possible. There are many hosting plans available for under $10 that allow you to pay monthly. You will need to pay 1 year of registration fees to transfer your domain however, but then you're domain won't expire for another year.
The real question is what are you looking for in terms of space, bandwidth and other features.
I would recommend the host I use for lots of reason, ashosted.com , dont know if im allowed to post the url so delete if not, but I have used them for 4 years now, great company.
Sure - however, you might want to leave your domain where it is and have them just move the *files* from your previous hosting account to your new account.
I'd recommend you keep your domain with a well known registar (namecheap.com and godaddy.com are two) and simply point the domain to your new host by changing the nameservers associated with your domain.
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It is definately possible. If you still have a year before your domain expires and it is hosted with a reliable domain host I would suggest leaving it where it is for now and just changing your Name Servers to point to your new website host.
This will save you the transfer fee for a while and also give you a chance to see how reliable your new host is before you transfer your domain name.
yup dont go transferring your domain to a host unless you are sure about them.. and if you have to transfer your domain then do it to a well known registrar...
It's possible to find decent webhosts at that budget. I think you should keep your domain with a domain registrar like GoDaddy or NameCheap.
I would echo again what the other posters have said. moving your domain name to a dodgy registair can leave you in limbo. for example one registar saying you have brought the domain and is pointing to a server and godaddy saying its available!!!