Why not 'mającego'? Isn't it genitive in this case because of the negative?
"Mając" is an adverbial participle and refers to the subject which is in Nominative.
Sooo why punktów?
Because of "pięć".
Is it possible to create polish sentence which doesn't use "having"(mając), but "with"(z or something)?
You could say "Nie możecie wygrać z tylko pięcioma punktami", although it doesn't sound great to me.