So I haven't updated my xbox in over a year. (I forgot that keeping this thing up to date is a full time job)
I've got a Hitatchi drive from an original xbox with ixtreme 1.51 updated on 7/8/2010.

So please help me understand what I need to do:

1. Stay off line?

2. Update to newest firmware? Which one? Any tips?

3. Download a TU file from http://www.xbuc.net/ and put it on a usb drive in order to play any newer games? or something??

Bonus: Explain something about lack of AP2.5 on my drive?

None of the above. You have a Hitachi drive, which:

A. Can go on Xbox Live without being detected, so don't worry about XBL
B. Has new firmware coming out, but it's not out yet so sit tight.
C. Just like the Samsung MS28 does not have ap2.5 checks so you don't need to worry about that, when the new firmware comes out you will be able to flash it then play whatever you want.

Cool. So I should just be able to burn my backup with burner max and be good to go?

I don't have to worry about the silet dae update here?
http://www.xbox360iso.com/silent-xbl-updates-t510556.html

And there's no benefit of updating to LT 1.1?

(I only play originals on live anyway but I want to take every precaution against getting banned so I don't have to have two consoles.)

Yes but until the new f.w for your drive (and mine) comes out you won't be able to play XGD3 backups, everything else will play fine. You don't have to worry about the silent update because your drive does not have ap2.5 checks.

You can also play backups on live with your Hitachi, I've been doing it for a loooong time now and no ban.

To be honest I don't know what is updated in LT 1.1 compared to iX 1.51, but the new f.w is in final testing so it shouldn't be too long before it's released.