First of all Kane won't be joining PSG according to most recent reports. Because he rejected PSG despite a really big offer made by them. But as for Vlahovic yeah he is very likely to join PSG soon. As for your concern about Mbappe situation PSG are really not preparing themselves for Mbappe's departure yet.
They actually don't want to let him go for free but there won't be any contract extension as Mbappe stated. In this situation I would have put him on the transfer list already if I was in PSG's shoes. But for now there is no update on this situation. I started to feel like Mbappe will stay this season as well and as of next summer PSG will start to be in search of making another sensational transfer like Mbappe.
It's true what you said, Kane is not interested in joining PSG, in other words, he refused and now Bayern Munchen is the club that is at the forefront of getting the services of the England National Team striker. Since the transfer market began, Kane name has often been associated with big clubs, PSG, Man United, Madrid and Bayern Munchen are reportedly interested in him. Now, with this statement, Kane future can be guessed. Join Bayern Munchen or stay at Tottenham for another year.
I think the most plausible reason now for Kane to leave immediately is that Tottenham don't play in the Champion League next season. My prediction is that Kane will move at the start of this season, even if Tottenham try to keep him but playing in the Champions League is the biggest desire for every player, thus his place in the national team will remain secure.