mattj and I tested the following this evening:
Balloon: is Balloon's effect nulled under the effects of Gravity?
Yes. The Balloon will remain until it gets hit, but it won't provide a ground immunity while Gravity is in effect.
Body Purge: Can its weight reduction be Baton Passed?
It can't.
In our test, we had Smeargle and max HP max defense Chople Steelix (400 kg) vs. Toxicroak. Smeargle used Body Purge and Baton Passed to Steelix, and Toxicroak used Low Kick on Steelix, doing 90 damage.
The normal damage range is 81-96 HP, and the damage range from halving weight would be 67-81 HP. The data entry says that 200 kg, half Steelix's weight, is hit for 100 base power instead of 120.
Analyze: What is the power boost?
It's almost certainly 1.3x.
In this test, we had a hacked min speed Analyze Magnezone use Thunderbolt vs. max HP Blissey. The normal damage range would be 132-156 HP without Analyze.
With Analyze, the damage was:
187, 174 (crit), 199, 183, 181, 190, 193, 195, 183 (crit), 199, 199, 187, 174
[or a range of 174-199]
At least we know for sure it's between 1.25x and 1.33x, because the highest damage from 1.25x would be 195 (too low), and the lowest damage from 1.33x would be 175 (too high).
With 1.3x, the damage range would be 171-202 HP.