Of course jalbreaking is not always used for illegal purposes. Most people only jailbreak for the unlock. I personally have always jailbroke my iPhones for the increased functionality, mainly themes. It's the single greatest thing you could possibly do with the iPhone.
As for installous. Yes you can download apps that are no longer in the app store but theres a reason why they have been removed, only apple knows why. Also how do you pay for the apps you download from there? If you downloaded an app from the app store then since deleted it form your phone and wish to put it back on, but find it's no longer available. All you need to do is check for available downloads in iTunes. If it gets removed from the app store before you get a chance to buy it then it's tuff. No need to go steal it.
As for app sync. It let's you sync apps that have been cracked. These are illegal even if you own the original. Someone has modified a programme without the programers permission and distributed it. Thats illegal. If you want a copy of your app backed up on your computer use iTunes. That's what it's there for. Failing that if you do delete an app and wish to reinstall it you can download it again for free anyway...
*edit* As for your query,
Usually when you sync, iTunes asks if you want to transfer the purchases installed on your phone to your library. If you choose not transfer, it wipes the new purchases you've installed. What you've probably done is tick the "Don't ask me again" box and put don't transfer, so every time you sync, if that app isn't in your library, it wipes it off your phone...
To fix this, when your phone is connected, right click it in iTunes and there should be a "Reset Warnings" option, or something like it. Do that then the next time you sync, iTunes will ask you to transfer purchases you made. It should work fine after that.noting doesn't it's because you've installed the apps from installous...
