Can "tar" backup incrementally?
What you describe is very similar to the way backup2l operates. You declare a home directory to backup and the 'backup scheme' (basically, the levels of backups).
The end result is a set of tar.gz files with incremental backups. You don't even need the backup2l tool to restore files (although is very convenient).
Optionally it can generate md5 checksums (option -m).
backup2l is a bash script, so you can make it fit your needs.
Yes you can, here is a tutorial to do just that.