TLV 3 night stop over on a round-the-world tickets - ideas

Ideas -

Rent-a-car ?

Public Transportation ?

Two gay friends travelling together ?

Tours ?

Budget travel but limited to a budget if worth it..

Ideas -
Perhaps you can provide more details about what you are interested in - history, religion, nature, food, culture, etc, etc. You need a minimum of 10 days just to scratch the surface of Israel, so for a three day visit you really have to define your priorities !!

You have time for a short visit to 3 places: Jerusalem, the Dead Sea and Tel Aviv. You could spend a few days in each but you don't have time for that.
For the 2 cities, a car would only limit you. As for the Dead Sea- there is an option to do it as a day trip both from TLV and Jlem and I strongly suggest a rental for the day.
No problem at all for a gay couple (obviously common sense and modest behavior in places of worship in Jlem). You'll find a very lively gay scene in TLV.
If you have 2 nights, I would spend 1 in Jlem and the other in TLV.
Suggested plan for 3 days/2 nights:
1. Train to Tel Aviv. Spend the day, go out at night. Sleep there.
2. Early morning bus to Jle. Spend day there. Sleep there. Rent a car in the evening.
3. Take luggage with you. Drive to the Dead Sea. Spend half the time in Masada and half on the beach. Drive straight to the airport. Return car there.

3 days is really nothing, especially if you are after a long flight, or before a long flight, or both...

I'd stay in Tel Aviv, which is the best place to chill. And I'd take a day trip to Jerusalem. If you're intent on cramming in as much as you can in 3 days, then you can consider hopping to Haifa/Akko for a day, and Masada and the Dead Sea are also a possibility for a day trip from Jerusalem.

I'd chill out in Tel Aviv - lots of gay-friendly spots - and do a one-day trip to Jerusalem's Old City and possibly Bethleem.