Is it possible to browse my iTunes backup files?
Yes! It is possible to view the contents of backups. There are several programs you can use, the one I use is iExplorer v3.

You can view and extract:


Messages
Contacts
Voicemail (though I have not tried this)
Call history
Notes
Safari (bookmarks and history)
Calendar
More if you are willing to dig into folders


The free version allows you to view the contents, but has restrictions on extracting them.

There is also iPhone Backup Extractor which does:


Photos
Recordings
Again, perhaps more


This is also a free / paid app, so have a look at what you need.

Hope this helps.