No, young Sigma got sent back into his body shortly after the events that took place in ZTD. The only version of him that winds up knowing those events is the Sigma that fails to prevent VLR's events, by which point he's sent back to that future and it becomes too late for him to do anything. Sigma was always going to go into the Mars testing site as an old mind with little information on what took place there.
I'll have to concede in this one because I can't remember that part very well, but I'm very convinced that sending young Sigma was the only way to prevent radical 6 from escaping because old Sigma knew exactly what happened (Sigma himself prepared the events of VLR because of this outcome) and thus can't pull off a schrodinger cat.
Alice in VLR was able to conveniently jump when you choose Ally or Betray, as was the case with Junpei.
No, it was never stated that Alice or Tenmyouji jumped, in the case of Alice at least (I don't recall Tenmyouji's part), it was explained almost as a coincidence as Alice in both cases explained why what she choose was the the smarter decision without even mentioning something like "because you choose ally/betray before" like Sigma and Phi (who are the only that knew what was happening) did.
They also explicitly remind you in VLR that exposure to other Espers for long enough allows you to gain the ability yourself, which was the entire point for Sigma and Phi.
Wrong, they say that espers get stronger (as being better at what they do, not gain more powers) when they are grouped together and isn't even hinted that normal people can gain esper powers by staying around them. If that were the case then why would Alice need Clover anymore if she could just gain the powers herself?
So why is it an asspull that Carlos and Junpei also end up jumping when exposed to the same conditions? (Note that Junpei and Carlos have shown signs of esper abilities before, while both young Sigma and Phi initially did not)
Because, while is true that is explained that Carlos has the same esper powers as Sigma and Phi, they had to spend a whole nonary game to even be able to remember part of what happened in previous timelines while Carlos was able to do the same in a matter of minutes and without losing any memory.
Junpei is a different kind of esper, he can access the morphogenetic field and store information there to use later. In ZTD they explicitly say that Junpei is jumping between timelines the same way Sigma, Phi and Carlos do, not that he is creating a backup of his memories and restoring them in another timeline which could or not be possible, nobody knows the limits of those kind of esper.
Also, why is Akane grouped with that question when we know her esper power is quite, well, powerful?
Same as Junpei, she is a different kind of esper and that is clearly stated in VLR, Sigma (IIRC) asked her why she needed him if she was an esper herself and that was part of the answer, the other part was that she knew what happened so she couldn't pull off a schrodinger cat.