I'm pretty sure it's correct, you don't need 在 / 里 here
I have a feeling this is incorrect: 在...里.. sounds like an expression of place, not time. What do you think?
It could be said.. "我在寒假里玩游戏" but agree this just sounds weird.
Li is not necessary
Otherwise＂你想在寒假里做什么？＂ should be correct.
The chinese is lacking
I am relatively certain that this should be accepted 寒假的时候你想做什么 and also probably this 在寒假你想要做什么. Any reason why they wouldn't be?