I'm FODMAP and Nightshade sensitive along with a whole bunch of other things!!
Definitely cooked veggies are easier to digest and highly recommended for those who have issues. Some fruits they recommend removing the peel (ie: apples) for those of us with compromised digestive tracks. (GAPS recommends applesauce for tummy issues but to remove all peels before cooking them)
Raw foods are much much harder to digest and will cause more strain on your system to break it down, creating more gas; you'll feel the results of this as your system is working harder.
Bonus question: not a scientific answer but compare a raw and cooked carrot, the cooked carrot is very tender and sweet and hardly needs chewing to swallow it. Once this hits your digestive system it is in a much better state to digest and doesn't demand as much effort for your system as a raw one would.
Starches, when cooked, become gelatanized which makes them easier to digest. Put cooked starched back in the fridge and you'll end up converting some of that gelatanized starch into resistant starch again. Resistant starch isn't a FODMAP per se but it is a soluble fiber that is fermented in the colon. I think FODMAPs are defined by being more quickly fermented (and, if I recall correctly, fermented higher up in the digestive tract). In any case, if you are having problems with FODMAPS cooking starch should only help.
As with actual FODMAPS, I don't know of any reason why cooking matters, but would be curious as well if anyone can chime in.
Cooked vegetables are easier to digest, reducing the potential of fermentation in the gut. I currently stay away from all fruit and honey since there's not much to be done about the fructose content. Fructose is a FODMAP too.