Mac – Why won’t Time Machine work with a drive connected to an Airport Extreme

As above the Time Capsule is the easy solution.

I can only really comment on your second point, I bought one about 3 years ago and 18 months ago the hard drive failed, it seemed to be a known problem as Apple replaced the Time Capsule free of charge even though it was outside of the warranty period. What could not be replaced however was the 18 months of backups as Apple would not have replaced the Time Capsule had I opened it up to get a the drive and try to recover the data. So after that experience I won't be buying another Time Capsule.

If you are not particularly after a router there are now other companies that offer Time Machine compatible network storage, which may be worth looking at.

I have run TM backups both locally and across the network.

Moving backups:

Just moving the disk won't work. If you don't need to use the old backups, it's probably worth just biting the bullet and waiting for it to complete a new backup (if you connect an ethernet cable on a gigabit network it's faster (~90MB/s) than connecting the drive via USB (~30MB/s)).

If you need to keep the old backups, you could try creating a sparsebundle with disk utility, mounting it, and then copying the backup files directly onto it. This method would be slower than creating a new backup from scratch over gigabit though, so if you don't need the old backups, it doesn't buy you anything.

Multiple machines:

Yes, you can backup multiple machines to the same Time Machine disk