Are deletion options available from within the Time Machine app itself safe to use?
Unless you are running very old operating system on a Mac, this option has not been available within the Time Machine application for a long time. Reason.....high probability of corruption of the entire backup file.
Is it safe to delete backups via the Finder? I am hoping that the Finder is smart enough to manage the Time Machine "hard links."
Not in my opinion, but it does work for some users and not others. So, there are no guarantees here. For what it's worth, Apple does not support this in any documentation that I've ever been able to find. The "genius" types at the Apple Store here tell me simply......."Don't do this."
My impression is that you can't delete anything from within the Time Machine app. Is this correct?
Correct, unless you are using a Mac with a very old operating system. Even then, there are risks. I've lost entire backup files in the past trying to do delete old backup files.
Is it safe to open a sparsebundle via the DiskImageMounter and then delete a backup via the Finder?
Not in my opinion. Again, this is not supported by Apple, so there are no guarantees.
Your best.....and safest bet, no matter whether you are using a Time Capsule or external drive.....Keep a copy of the old backups around for a few months if you think you might need to go back and pick up a file.
Connect a USB drive to the USB port on the Time Capsule and use the Archive function in AirPort Utility to copy everything on the Time Capsule drive over to the USB drive. Then, erase the Time Capsule drive using AirPort Utility and start over again with a new complete backup and move forward.
Keep the old external drive around for safe keeping. Add a new external drive and start a new complete backup to the drive and move forward.
After a few months, you will probably not need the old backups and you can erase the drives and use them as spares, or set up backups using Carbon Copy Cloner for a dual backup strategy.
Bottom line.....it is best to leave the existing backup file(s) alone and not try to modify them in any way, unless you have another copy of the backups handy......just in case.
It might be another discussion, but I've never been able to figure out why users want to keep backups going back many months, even years in time when they will never need them.