Regarding Mac Time Machine - is the size of backup dependent on the amount of space USED on the disc to be backed up, or is it dependent on the size of the disc, regardless of whether space is used or not.

Time machine will attempt to back up your entire computer (less the System Folder on recent Macs) onto whatever drive you provide. Any drive is SO much better than no drive at all.

Time machine First makes a complete copy of everything. Then, over time, backups are incremental -- Time machine saves files that have changed. Identical copies of the SAME files are not retained, which keeps the size from getting out of hand. But time machine has the ability to restore any file (or your complete Mac) still in its backup set to any date you specify.

Old backups are removed when space gets tight. The larger the drive you provide, the farther back in time will be retained. A backup drive 2 to 3 times the size of your 'stuff' will suffice for now. There are a few diabolical cases where an old complete backup and a new complete back must 'pass each other'.

For long term trouble free operation, babowa has it right -- a drive 2 to 3 times the size of the sum of your drive(s) is best.

Time machine works at low priority in the background, so Time machine does not get any extra benefit from a drive that is FAST.

Apple's recommendation for a TM backup drive is 2 - 3 times the total size of your hard drive (which will be 4 TB).

Time Machine automatically makes room for data when storage is short. It will delete older backup data to make room. So when Time Machine sees that that 4TB LaCie is getting close to full it will start deleting old data.