IPhone – Restore deleted iTunes backup of iPhone

It depends whether the remaining files you have are from an encrypted backup.

It won't be feasible to read your backup data if you have an encrypted backup with no Info.plist, Status.plist, or Manifest.mbdx files

However, if your backup wasn't encrypted, it will be possible to read that data. You'll find that whilst the filenames are unhelpful (they're SHA-1 hashes of the file's internal path), the file contents easily readable. It's only the filenames that'll be lost.

If you're looking for images, try renaming a bunch of the files to have .JPG extensions, and see whether Windows Explorer or macOS Finder previews them for you.

There are a few approaches for digging into what's there in bulk. You mentioned using iPhone Backup Extractor, and there's a guide on how a use "Bulk Rename Utility" to read backup data when in this situation.

No. iTunes performs two different types of backups. One is used to carry over information during a major update (as in the move to iOS 4 to iOS 5). This backup is only stored and used for this brief period of time; think of it as a temporary backup to serve a singular purpose.

The other is a permanent backup that can be deployed so long as you hold onto it. These backups are listed in the iTunes preference window:

As for trying to recover the backup, there is no way to do this because frankly, there are no files to recover. Even trying to use 3rd party disk recovery utilities on permanent backups would be futile because the backups aren't stored in a singular file, but rather littered through a thousand of them, each uniquely carrying an identifier. Something a recovery suite could never effectively retrieve.

Sadly, you have lost it all. The only option would be to jailbreak and perhaps run a recovery utility on your phone's filesystem. I've never done or heard of it being done, so you would likely be in uncharted waters.

For the future, it's always a good idea to right click on your device as it's listed in the iTunes sidebar, right click and selected Backup from the context menu before any major update. Additionally, iTunes will create its own permanent backups periodically, so it's also a good idea to sync more often than you've been syncing your device.