In general, how can we build the passive forms from verbs which end in 하다 and 주다? Can we always use 되다 as passive of 하다? E.g. 시작하다-시작되다 or 청소하다-청소되다?
What will be the passive of 영향 주다? Can we use 영향 받다?
We can use 시작되다, 청소되다, and 영향을 받다, but in general, you have to memorize passive forms one by one. And many have restricted usage - I don't think you can change "어젯밤 그 문제를 생각했다" into passive, for example.
In Korean, a "passive form" is best considered a morphological derivation (creation of a new word), while in English it's an inflection (different form of the same word).
In English, for example, you can almost always turn a verb to passive (by inflection), and the rule is mostly regular. On the other hand, if you want to turn a verb to a noun (derivation), then you have to memorize them one by one, because they are actually separate words. E.g., you must say "arrival", "departure", or "start", and not "arrivure" or "startal". Some like "hear" doesn't even have a noun form.
Korean passives are basically like that.