Apps will be restored from iTunes when you put in the same iTunes ID as before. The data that goes with them is a problem, though, which is one reason I've never done it this way.* Some of those apps (like Splash ID, ToDo, DocumentsToGo, and others) have their own sync mechanisms, so the data for those would be okay. But if there some apps with important data and no way to sync or export from them, then saving
that data would be very difficult or impossible with this kind of restore.
*Well, I've done it once to prove to an Apple "genius" that a problem I had was hardware related and not software related, but then as soon as I brought the new hardware home I restored from a backup.
My guess is that the garbled search results problem is not related to corrupt data anyway, so there is likely no point in trying this. (The problem the other person was having w.r.t. the calendar not even starting up properly likely
was corrupt data in the iPhone's database, which is why restoring from it restored the problem.)
Just keep sending Apple feedback. It takes time, but they have eventually fixed nearly every bug I have reported that way over the last couple of years.