I generally register the domains (with godaddy) initially and bill for them along with the site build.
When I'm setting up email accounts (usually google apps) I move the domains into their own account (which is free and quick with gd!) so the client gets the reminder emails and the like. So long as you let them know to look out for the mails, and whitelist the gd email address then all is fine.
I usually get a call sometime afterwards going 'I got this email...' had one this morning in fact, but a least the client pays with their own cards and they are responsible for ignoring the emails.
Hosting wise I have an account with servage.net (for sites with <1000 visits p/m) and I offer that for a nominal charge of 40 GBP as a one off. I have several personal sites on it so it's no real cost to add a few low/medium traffic sites on top.
For clients with higher bandwidth needs (>1 million visits p/m) I have a fully loaded dv with mediatemple which can be shifted to and from different plans according to bandwidth needs (at impressively short notice).
Their Plesk offers the ability to devolve admin to the client if needed and I bill this annually (using a google calendar for reminders) and issue catch-up invoices if I have to switch it up to handle traffic spikes.
I'm full time consultant, so I don't webmaster as a profession. The sites I build and run are generally for friend's small businesses/charities/hobbies and the like, all of whom asked for help because they had been ripped off elsewhere. I don't make much out of it but I do learn a lot and that helps with my day job. The scale of my 'problem' is about 30 sites, with, in total, <6 million uniques a month, I imagine the world would would be very different further north of 60 sites.
Do I recommend these companies?
Servage is reliable for http, their customized control panel is very much easier to use than cPanel and makes managing DNS a breeze. The support leaves a lot to be desired as does sql availability. You get what you pay for! and servage offer unlimited everything for 80 GBP a year.
Media Temple are awesome! Their support are very helpful on the phone. Considering the server is west coast usa the roundtrip for uk sites is impressive, as is the ability to switch to a higher bandwidth balanced infrastructure in a phone call.
Godaddy are cheap and have a deliberately confusing website, but their domain manager is very good, phone support have always helped me out when I needed it (and being american based very politely), and being able to ship domains into new accounts is a great feature I haven't found elsewhere.