How exactly is the probability of at least one event happening dependent on the probability of all events happening? Say we have $P(A) = 0.60,\, P(B) = 0.50$ .

Normally to find the probability of at least one happening, we find the probability of neither of them happening:

$$P(A^c \cap B^c) = 0.40 \times 0.50 = 0.20$$

and then subtract it from $1$ ( getting $0.80$). I know that this is incorrect because we are also given that $P(A \cap B) = 0.35$, thus $A,B$ are not independent. How would one go about finding the probability of at least one occurring in this case?
The event $C=$"at least one from A or B happened" is the same as the event $C=A\cup B$.

So.. just use one of the most basic results in probabilities: $$P(A\cup B)=P(A)+P(B)-P(A\cap B)=0.5+0.6-0.35=0.75.$$
Hint: The probability of A happening and B not happening is $0.6-0.35=0.25$.