These are all good questions! Often times, when solos are given in these kinds of creative contexts, the soloists can make up music on the spot. There's an understanding between the lead musician (here, Yanni) and the invited soloist that whatever they make up would still be stylistically appropriate while they are exercising a certain amount of creative freedom. So, while it's possible that Yanni is writing down every single note of the solo and the player is just playing that, I would guess that it's more likely they are improvising.

As for choosing to use the conductor -- That will depend on not just the piece of music (is it complex?), but also whether the players have enough rehearsal time to do it without the help of the conductor. Sometimes, the conductor is the lead personality, so it's going to start with that person and even if the music is not super complex or needs the conductor to keep everything together, the vision is the conductor's, so the ensemble will feature that person. It just all kind of depends.

Good questions.