For some reasons, red papaya I bought ripen very slowly (like 2 weeks or
more) at my house. I even put it inside a cardboard box with a bunch of
bananas. If this accelerates the ripening process, it is not very
significant. Well, the bottom side of the fruit is more ripe (yellow) than
the rest. Don't know if it has to do with the bananas.

The problem is not just I'm impatient, but the fruit also tends to decay or
grow fungus after 2 weeks. So if it is not ripen by then, I have to eat it
half-raw.

Does the room temperature and humidity have to be in a certain range?

i think the papaya you bought was subjected to chilling injury. Papaya
is climacteric and climacteric fruits are usually commercially ripened
by ethylene exposure or just leave them in room temperature.You said
that you have put the papaya with a bunch of bananas, so the typical
thing that should happen is it ripens in no time.

Chilling injury can happen when fruits are stored in temperature well
below their physiological tolerance. Chilling injuries usually result
in brown patches and failure in ripening.