Good questions!
It depends on what you're talking about. These days (since late 2009), music bought from iTunes does not have DRM, meaning you can buy a track and it will play on anyone's computer, regardless of whose account bought it, and using any software (as long as it plays AAC files).
Movies and TV shows still have DRM, so will only play on a computer using iTunes, or an iPod device, under your iTunes account. Same goes for apps. Although you can have up to five computers authorised at any one time, and any number of devices syncing with those computers.
Playback of music will continue to work regardless of whether Apple exists or not, as there is no DRM. TV and movies (and apps) will continue to work too, but only on computers already set up with your iTunes account - if Apple's servers disappeared tomorrow, you'd have no way of authorising a new computer on your account, only existing ones would still work. You'd be able to sync a new (to you) iPod device, though. Note that I'm excluding rentals for the obvious reason that rentals WILL stop working after the rental period expires, as that's the whole point!
One music service - I forget which - did leave users high and dry when they turned off their licence servers, and any tracks users had bought became useless...
To clarify the matter with apps, at the moment the only way an app will stop working is if It's not compatible with your device's OS. For example, if an app has been written in such a way that it will only work on iOS 4, and not 5, then you could be forced to buy the new version when you upgrade your iOS device. Otherwise, an app will continue to work- but there's nothing stopping a developer from bringing out "My App", and then "My App 2" as a separate app. Most devs launch an app and then update it over time (such updates are always free), but it's not unheard of for someone to decide that their latest work is more like a whole new app, and so sell it as a whole new app. If that sounds sucky, don't forget that that's how it's been with computers for years- just because an app worked with Windows XP doesn't mean it will work with Windows 7. And Autoroute 98 isn't going to get free updates for ever... Oh, one last thing- all the App stores (not just Apple's) have a "remote kill switch" which, in theory, could allow them to remove an app from devices- but it's a last resort kinda thing, like if an app was discovered to actually be malicious. But a dev couldn't say to Apple "please wipe my app from everyone's iPhone so they have to buy the new version".
As for changing computer, you can only re-download apps for free. Music, films and TV are a one-time only deal; if your computer is nicked, and you haven't backed up your iTunes library, you'll have to buy them all again. Apparently in a very small number of cases, Apple have allowed people to re-download stuff, but it's certainly not the norm. Look at it is way, if someone nicked your CD collection, you wouldn't just waltz back into HMV and help yourself to new CDs, would you?
However, you can move your library between machines; so when you replace your computer (even if you go from a pc to a Mac) simply copy your iTunes library to the new one and off you go.
Hope that helps!
