You wouldn't be able to let them register the domain but yet still configure the DNS for them (unless they given you full access to their accounts, which would obviously be a bad idea).
There's nothing too wrong with registering client's domains for them, especially if you make clear that they own the domain (not you). And, if you really want to put your mind at rest, set your client's details as for "Registrant" of the domain (i.e. so when doing a WHOIS, it'll show that your clients own the domain, and you simply maintain it for me).
As for handling the host, WHMCS is a great billing (and support) software for hosting. It's a complete package, and very powerful. You have to pay for a license, although that's only ~$12 per month (or $5 per month with a WHMCS reseller).
Best of luck to you 
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