Because this long post for me to reply to, I'm going to do something a little bit different and use some dictation software to write it. I'll try and watch for bizarre mistakes, but hopefully this should keep the normal level of dyslexic errors to extreme minimum.
First, the proper terminology is outgassing. You reach into a solvent, like water, and outgas into an atmosphere. So technically your step dad is right there is no leaching occurring, unless you happen to store your figures in water. However he's wrong about the materials. Plasticizers do not bond to the plastic to become part of the polymer chain. If they did the result would be a copolymer. Plasticizers are dispersants in the matrix of plastic. It's more analogous to an alloy where you have molecules plasticizer trapped by chains of polymer. Also the US and EU aren't talking about banning Phthalates, a whole class of plasticizers, because they're staying put in plastic.
Second, I still think what I'm saying about plasticizer loss is still not coming across well. You have to lose a lot of plasticizers before you start seeing effects due to loss of the material. With vintage figures we are still not seeing any major negative facts on the plastic itself due to plasticizer outgassing. So I highly doubt you see any noticeable difference with your pressure test. What I am saying, is that a residue that is formed on the surface of the figure is evidence of some type of outgassing. The most likely additive that I can think of that would out gas is the most volatile. The most volatile additive that I can think of would be the plasticizer.
A tacky figure itself is not necessarily a sign of plastic breakdown, but it shouldn't be ignored. In the case of the battle droids I find it to be slightly disturbing because I know these figures were not tacky, sticky, or oily when I bought them. It's a change that is occurred while I have owned them. Something is leaving this sticky residue behind. Additives are in the plastic for a reason, into something is outgassing it is no longer serving its purpose in the plastic. The long-term prospects of storing plastics for say 50 to 100 years are what I'm concerned about when I talk about plasticizer loss.
I assumed Lego bricks didn't have a plasticizer because of the polybutadiene. Polybutadiene is an artificial rubber and has many of the characteristics that a plasticizer would have. I spent 3 and half hours searching for what additives Lego adds to it it's ABS. But couldn't find anything beyond the conform to government safety standards. So I'll stick by my original assumption unless you come up with anything to the contrary.
So back to the discoloration, your step dad is absolutely right about white being very difficult to get right and keep consistent. I think it's a little hard to absolutely rule out UV exposure unless you've kept your figures in a sealed black container. Many households light sources are capable of putting out small amounts of UV. Also photochemical reactions can happen with lower energy visible light. However in this case I think oxidation is more likely cause. Given what I currently know about the flame retardants, I would say they are your most likely culprit.
Well I hope I've caught most of the mistakes, but a long post like this and I'm likely to have missed a few.