I have a Motobecane Le Champion with down tube shifters:

1. Do you switch hands when you are shifting front vs rear derailleurs?

I use my right hand for both

2. Do you place your fingers on the down tube to stabilize your hand when shifting?

Sometimes yes, sometimes no

3. Do you use your thumb and index finger to push/pull the shift lever forward and backward?

Yes
4. How to you shift and brake at the same time?

If I have to do that, then I'm in the drops but I can only brake with the front wheel. I'm not a racer, just a tourist. I never race anyone. Well.... there was this really cute girl once 35 years ago wearing hot pants that I did my best to catch up to, but she got away.....

5. What else do I need to know technique-wise?

Just practice. I have extra problems. I have a 15-26 rear and a TA 26, 52 front. I have to be careful that the rear is not 15 when I shift up from 26 to 52, or the chain will drop off on to the right crank arm. And I have to avoid shifting back into the the back 26 when I'm already on the front 26---FAST. I shift into it, and then back quick. Or I could put the chain in between the back 26 and the spokes(Ok I final;ly bought one of those ugly metal guards because I got tired of chewing up spokes.)