Learn Finnish: Se oli hauska sattumus. - It was a funny coincidence.

Sattumus is derived directly from sattua (to happen by chance). While sattua describes the action of happening to do something, adding the -mus suffix turns it into a noun describing the event itself: an incident, occurrence, or coincidence.

I've also seen the word sattuma. Is there a difference between sattuma and sattumus?

Yes, there is a helpful nuance! Sattuma usually refers to "chance" or "coincidence" as an abstract concept, like fate or probability. Sattumus, on the other hand, refers to a specific, concrete episode that happened to occur. So hauska sattumus implies a funny incident or anecdote.

Since sattua can also mean "to hurt", does sattumus ever refer to an injury or something painful?

No, it doesn't. Even though the verb sattua has a dual meaning ("to happen" and "to hurt"), the noun sattumus only inherits the "happening" meaning. It strictly means a chance incident or occurrence.

If I wanted to express this exact thought using the verb sattua instead of a noun, how would I do it?