Can I back up both my iMac and MacBookair to the same LaCie external drive recently bought from Apple and using Time Machine

You can but that's a bad practice as drives fail, not partitions. If the drive were to go bad, you'd lose everything back-up from both computers.

TM backup HDDS should be dedicated and 2-3x the size of the internal drive, i.e. each computer with its own backup drive.

+1 A very poor idea to try and save money. Each computer should have it's own External Hard Disk for backing up to! If you execute your plan, it is not a matter of if you will lose all of the data on it, it is a matter of when!

TM is always installed, it's part of the OS.

When our Notebooks sit on a desk, the backup drives are always attached. If not, you're not backing up. You can do 100% manually, but we just plug 'em up and let 'em run.

All things related to a computer require electrical power.

If your battery has enough charge for the period you plan on leaving it alone then no. If you have any expectation of the battery ever running down while you're away then yes. It's up to you. How do you wish to do this is what matters.

And no ... leaving the Mac "on the charger" all the time will do it no harm.

+1 (not a good idea).

But, if you feel that you must, do partition the drive first so each backup has its own dedicated space and the files won't be "jumbled" together.

the connection between the hard drive and the computer would be more stable if the external was supplying its own power source instead of relying on the USB port.