You're right, Haaland still has to prove his performance in the remaining 37 EPL matches. In the next match Manchester City will face Bournemouth and Bournemouth itself is a very weak team IMO. It won't be difficult for Haaland to score at least 1 goal here (if he is played early in the game).
At least Haaland has a good debut in Premier League and it gives him more confidence and energy for his adventure there. If Haaland can move and find space well, creative and smart midfielders of Manchester City can provide him a lot of key passes. So he won't face with lack of key passes to find goals. I think the biggest challenge for Haaland in Manchester City is not how many goals he can score for the club. It is how he will perform in big matches, in important ones that can decide Premier League title and Champions League title.
Regarding Mbappe, I don't think he has to move to the EPL to create a rivalry with Haaland, we'll see their rivalry in the UCL soon.
Mbappe has big love for Real Madrid so if he leaves PSG, he will move to Real Madrid and La Liga. If he moves to Real Madrid and get success there, it's hard to see Real Madrid let him go and join Premier League. I believe we will see Mbappe in La Liga but not in Premier League.