I had to come back online to set-up my Switch (which finally got here). I've played a little bit of Mario Odyssey, and slowly getting a hang of the controls (I still need to commit the dive actions to muscle memory).
I'll tackle these one at a time.
The thing about Gohan is that he doesn't really like to fight, so he doesn't train for fighting unless he needs to, at least not until after Resurrection F. Even if his Potential was unlocked, if he doesn't keep up with his training his body will weaken, and likely, so will his connection to his Ki. After Resurrection F, he realised that if he allowed himself to get too soft, then there might come a time when Goku and Vegeta aren't there, and he's the third strongest fighter, so what then? Now he trains to protect the things he cares about.
With Master Roshi, one of the things about the series is that, even when someone thinks they've reached their limit, they can always push a little harder (basically the running theme of the Tournament of Power). In Dragon Ball, Master Roshi thought he was as strong as he could be, but Goku, Krillin and Tien showed themselves to be able to surpass what Master Roshi thought was the limits of mortal men, so he started training in secret (Goku even says this in Super, I believe it was in the Yurin episode, something along the lines of "So you've been training in Secret, huh, Master Roshi?")
About training for the Androids, Piccolo, Krillin and Tien did get way stronger (Piccolo wouldn't have been able to deal with Dr. Gero like he did otherwise), they were probably stronger than Goku's base level (in theory), it's just that Goku also has Super Saiyan (a 50x multiplier) so Goku can become multiple times stronger in an instant.
Finally, regarding Piccolo, Tien and Yamcha not learning King Kai's techniques, they were only on King Kai's planet for half the time Goku was there (Goku was on the planet for a whole year, while I think Tien, who I believe was revived last, was only there for about 6 months, Piccolo was only there for about the time it took for Bulma, Krillin and Gohan to reach Namek, which I think was a few weeks). Piccolo barely even engaged with King Kai in the short time he was there, just meditating under the 10x Gravity inherent to the Planet. At best, Tien probably learned either the Kaioken or the Spirit Bomb, but not both, and has never actually needed to use the one he did learn; I'd assume he learned the Spirit Bomb, if he learned either at all.
Not refuting much because you're spot-on, except the fact that Goku was not there for a full year. The saiyans took a year to get to Earth, but Goku spent about five months with King Kai after having to travel Snake Way, and started when he had a power level of like... was it 350? I don't feel like looking it up. But due to training for the saiyans, the Z gang had become
much stronger than Goku was when he started down the Snake Way path, as King Kai said, so they already had a head start before they even started training under ten times normal gravity.
The other thing is that we actually have
no idea how strong Gero is. He didn't know about Namek and the super saiyan transformation, he only knew about Goku's fight against Vegeta. With that in mind, the highest power we can safely assume for Androids 19 and 20 are "higher than Kaioken x4 Goku when he fought Vegeta". It
could be that they're as strong as a super saiyan, since Gero clearly showed the capacity to do so with 16, 17 and 18, but we aren't shown a ton of evidence for anything significantly stronger. Goku was beating 19 very handily despite an oncoming heart attack, and Vegeta as a super saiyan-- who should be much less powerful than Goku at this point since their base powers were so vastly different before Vegeta died-- was so far above 19 that he let him suck a large portion of his ki before destroying him anyway.
What I'm getting at is that the three years training for the androids
did make everyone stronger, but we have absolutely no frame of reference for just
how strong because none of them fought the same enemy that Goku or Vegeta did in one on one combat except Piccolo fighting 17 (same as 18, close enough) after fusing with Kami and Tien stalling Cell with his Kikoho (which, if I remember correctly, was only like two or three shots in the manga and not the, what, fifteen? in the anime for filler). With that in mind, Piccolo beating Gero handily pre-Kami-fusion might not be as impressive as we assume. They can't feel Gero's ki so no one's able to confirm his actual power level; we just see him get walloped then one-shot-KO'd by 17 very effortlessly.
EDIT: Also, something that comes to mind that bothers me about Super's approach to "training" in general, just to throw this up for discussion.
I.
Hate. The idea of "I've been secretly training on my own" having any substantial effect on one's power. Dragonball has shown in action that greater heights in power are achieved by facing a stronger threat and rising to the occasion. One needs that opposition to fight against and grow. Roshi, 17 and Tien all "training offscreen" is such a cop out to me. Yes, Roshi could continue training in secret, neato. But he didn't fight Buu. Or Cell. Or the androids. Or Frieza. Or the Ginyu Force. Or even a freaking saiyan. What exactly has he done to make up the large gaps in power on his own without a training partner to work against? Fire Kamehamehas at the ocean? Same goes for 17; he's adamant about not disturbing the wildlife and he barely sees his sister anymore, so
who was he training with? The animals? He doesn't want to create craters or destroy trees, so where did he exhaust his power? What frame of reference does he have for even improving? Is he just working his mechanical parts with endless pushups?
In DB and DBZ, large growths in power are justified and explained onscreen via story elements and conflicts. Even when they're cop outs (hyperbolic time chamber, fusing, zenkai boost, etc.) they at least have a story explanation. "I've been training while you weren't looking," to me, is just incredibly lazy. And it's a large reason for why I don't like Super as much as most.