may I use my external back up drive from my previous Mac to back up my new Mac without losing old "back-ups"?

If you backup your new Mac, you'll have every thing on the Mac. If you're worried about history, its importance to you will determine what you do.

HDDs are cheap and reliable. If you wish to sequester the historical data, get a new HDD and start afresh keeping the old drive in a safe place. How often you've used historical data should help determine its usefulness.

Yes, but if the drive is more than 4-5 years old it would be very wise to get a new Time Machine drive. HD's wear out and this is a great opportunity to change it.