Just from the brief burst of SSJ2 power, Vegeta realized that Goku was still far stronger than him. Getting stronger wasn't the only reason he allowed Babidi to possess him but it's still a reason.
Vegeta doubted if he had surpassed Gohan till he realized how much weaker he had gotten. He was probably a fair bit weaker than Pre-Teen Gohan before the majin charm.
It's a bit more unclear with the anime due to all the padding but even then Vegeta, Trunks, and Piccolo were the only ones still standing in the end... indicating Piccolo is not too far behind them.
He looks a bit more hurt than Vegeta and Trunks in this cover, but still standing. Goku...
Vegeta and Trunks were brought up specifically because they are simply doing the best out of all the other Z-Warriors. Piccolo probably was getting his ass kicked as well, but not to the point of being instantly floored like the Earthlings and exhausted MSSJ Goku.
Yes, but it's very one-sided (though much less so in modern Dragon Ball). Vegeta was/is very obsessed with surpassing Goku while Goku being Goku, ain't really worried about Vegeta. He's just focused on surpassing himself.
Tbf, Goku specifically mentioned SSJ3 taking time he had left on Earth...
Did he really? All he said is that he had enough energy to destroy the solar system with his Kamehameha. He did say if he won the Cell Games, he will slowly and metaphologically kill everyone on Earth if he won. He likely would have repeated the process on other planets across the universe.
I...
Warm-up/Initial - The level he used against Vegeta and Trunks.
Buff - The level he used to show how useless Grade 3 is and his psychological breakdown against Gohan
Suppressed (Vs. Goku) - The first powered-up level he used against Goku. His speed is more suppressed than his strength...
Goku would be stronger than he was at the Cell Games since he wouldn't have to teach Trunks SSJ (he already unlocked it) and Trunks may take Gohan's place, while Gohan probably never achieves SSJ at all since Vegeta wouldn't even bother with him. I don't think he even actually trained Trunks...