The way OSM does it is hosting the tiles directly on an instance of Apache. The tiles are stored in the TMS fashion, rather than WMS. They have a custom Apache module called mod_tile which takes care of rendering queues (it’s a bit more complex than that but you get the gist).
Yes. There are planet files released on a regular basis (weekly). These are very large however. You can get extracts for many places and there are tools which work with OSM data and can produce extracts for your own specified polygon.
There are also hourly and minutely difference (diff) files released which can be applied against your database to keep the data fresh.
Why Flash? You’ll take a performance hit and the bandwidth usage will be atrocious (it will anyway, as mapserving is bandwith intensive but Flash files are typically pretty big and will cause more stress than is necessary).
I appreciate that you are/may be serving data as well as just map tiles however OpenLayers has vector drawing features which I expect will do what you want. Not sure exactly what features you’ll need though.