They will be told at least some basics on their powers! The Medbay does some powers scans while they're in Medbay at the start of their stay on Legion World. However, they don't know everything about the powers (for instance, if there's some quirk to activating them, they'd have to learn that through practice) and can only give the absolute basics.
And yes, it is the Time Trapper that gives the powers and the Time Trapper has at least some measure of control over what powers they get. The Legion has no say in what the Time Trapper does, nor do they get any warning or control over who gets what. The Time Trapper is acting independently and it bringing characters to the game universe is actually against the Legion's wishes. They ask the new people for help because they need it pretty desperately, but a lot of the Legion recruiting them is in the hopes that with their help they can resolve the whole situation and get them sent home. So they don't really have any say in who is grabbed and what powers the Time Trapper gives them, since it's doing it all against their wishes.
As for characters joining the Legion, they definitely don't have to do that right away. They're welcome to stay on Legion world where they have protection from some of the corrupt and evil parties in the game universe, but they can take some time to figure things out before actually joining the team. The only caveat is characters who haven't joined can't go on missions yet but they can experience plots that happen on Legion World or happen by surprise elsewhere. As an example, we had a character who didn't take the oath right away but after he helped protect the UP Council in a surprise attack, he decided he wanted to take the oath after. Joining the team can definitely be a process and the characters can stay on Legion World with all the other characters in the meantime.
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And yes, it is the Time Trapper that gives the powers and the Time Trapper has at least some measure of control over what powers they get. The Legion has no say in what the Time Trapper does, nor do they get any warning or control over who gets what. The Time Trapper is acting independently and it bringing characters to the game universe is actually against the Legion's wishes. They ask the new people for help because they need it pretty desperately, but a lot of the Legion recruiting them is in the hopes that with their help they can resolve the whole situation and get them sent home. So they don't really have any say in who is grabbed and what powers the Time Trapper gives them, since it's doing it all against their wishes.
As for characters joining the Legion, they definitely don't have to do that right away. They're welcome to stay on Legion world where they have protection from some of the corrupt and evil parties in the game universe, but they can take some time to figure things out before actually joining the team. The only caveat is characters who haven't joined can't go on missions yet but they can experience plots that happen on Legion World or happen by surprise elsewhere. As an example, we had a character who didn't take the oath right away but after he helped protect the UP Council in a surprise attack, he decided he wanted to take the oath after. Joining the team can definitely be a process and the characters can stay on Legion World with all the other characters in the meantime.
Does all this help?