... which would be 'How does the team feel?'

Yes, as the examples in the link show, 'What is the mood / atmosphere (like)' is a common option, but 'How is the mood' is also not wrong.

And yes, 'on the team' is probably the most common, or you could also say 'among the team.' 'In the team' could sound more Denglish, but you could say 'in the dressing room (BE) / locker room (AE),' or if it's not a sports team, you could say 'at the company,' 'among the employees,' 'in the department,' etc.

But if the question is mainly about which preposition to use with the word 'team'? Would we consider 'in,' or does it usually need to be 'on'?

It would indeed help to have a little more context from Claus II, including whether it's really a sports team, or just some other group of people.

Not to be a wet blanket, but ...

Doesn't 'the mood of the team' sound a little too artificial, like 'the pen of my aunt'?

I'm still inclined to go for 'the mood on the team,' but I could also be okay with 'the mood among the team members / players.'

This is one of those where the more times you read it, the more all the options sound strange. /-: