Does Time Machine do all I need? I use Time Machine for regular backups, and I have multiple HDDs for redundancy. This works fine for my computer, but the problem is that I have many files that do not reside in my computer and should be synced between these HDDs.

For example, photos from iPhoto: from time to time I have to free up space in my computer so the photos goes to the backup disks and I erase them from my computer. The mess increases with my wife doing the same thing with her Mac.

So what I'm looking for is a solution where I can tell my computer that I don't want certain files to be in it anymore, they should go to the backup, and that also the backup should be synced between the HDDs.

Is this possible to achieve using only Time Machine or any non-third party tool? If not, are there any third party solutions that do it?
Time Machine is a Backup Tool, not a File-Management Agent.

All it does, is copying new files to the HDD (as a mirror) and all older/deleted files go into a time-stamped archive.

If the disk is full, time machine deletes old stuff.

I assume you want to manage the data. Nit really "backup".
Because you do not copy them, you move them to the disk.

I also assume you have 2 Hard-Drives (1 for you and 1 for your wife), that should be in sync.

So in total you have 2 Macs and 2 Hard Drives.

Now Unix has a very popular command called 'rsync'.

You can sync (copy) or move data via rsync to the Hard Drive.

Rsync is command-line based, so it is a bit difficult.

But i would recommend it for your situation.

And because I was bored, here is a starting point for you.