Here's my advice and experience. Shared hosting is terrible once you reach a certain point. I've done some research into a company (http://servers.aplus.net/). A friend of mine uses their services and within the month, I'll be swapping to a dedicated box there. It starts as low as $99 for a box all of your own. Resets and things like that are included in all of the packages I've seen. I have, in the past, had the issues you speak of, but aplus spends a long time telling you that hardware resets and swapping out bad RAM will not only not cost you extra, but will usually be same day.
As for hosting from home/business. You'd need a fast connection. Not just fast download, like so many Broadband companies elate about. You need fast (or faster) upload speeds. Most standard ISPs will nail you to the wall for much use of your upload bandwidth, and you'll run out of your allotted very quickly. If you want to create the next-gen fad, you'll need something more than a home account can offer. You'll need anything from a Business Static IP account all the way up to a DS3 connection with the appropriate hardware to interface with it. I'd suggest by starting with a call to your intended ISP.
For your computer concerns, there are plenty of options. If the connection is solid, then the computer can start small and grow. You'll experience downtime if you want to upgrade and don't start with two load-balanced computers (so you could take one down for replacement/upgrade without the entire site going down). Don't expect to use that computer for anything personal either, not if you get much traffic.
As for power, a UPS can only go so far. You'd need UPS's for your comps, routers, and for your connection equipment. Most only last for 2 hours max. After that, gone. For a truly powerful option, you're looking pretty hefty construction costs. Co-lo facilities actually have more than one Internet provider connection and power company connection. They are usually wired in a configuration known as a sonnet ring. Ask your power company and/or ISP about it. They can give you estimates, they'll have a construction department.
Depending on your budget, I'd go with a start-small option of a basic third-party dedicated box. Keep everything backed up and upgrade later will be a snap.
