I wrote "trur" instead of "tror" and it was marked as also correct. What does "trur" mean? I tried finding in in several (online) dictionaries to no avail.
It's an alternative form of 'tror', which is less common.
Can we, in this case, omit the "at" as we would in English?
"Jeg tror han elsker meg" is that correct?
Yes, that would be fine.
Is the phrase "falling in love" recognised in norway, if so how would you say it?
I think it's 'å forelske seg', e.g. "jeg forelsker meg i deg". Might be wrong though :)