If I see an answer that involves some illegal activity, such as breaking license agreements, should I then mark this as Offensive or should I simply down-vote it?
I would be very conservative with that, as not all people on SE are from the same country and laws can be very diverse. I would comment on the answer that it breaks these [laws] in [country].
There are some perfectly legal things which people may rightly be offended by, and totally illegal things which offend very few.
It may sound trite, but I'd mark it as "offensive" if it offended me. If it didn't, I wouldn't.
It doesn't do any harm to let someone know that their post - even though not illegal - causes offense to others.
You mean, like an answer that suggests downloading a piece of "abandonware" -- something that is copyrighted, but no longer legally available?
Or discussions of breaking DRM, conscientious objection, and the like?
There are grey areas.
Unless it's really, really, really blatant that what they're doing is illegal, I'd say no.
Why would you want to mark it as something other than what it is? If it's a concern, add an option to mark it "Involves illegal activity".
Copyright circumvention is a grey area, as many nations' copyright laws are ridiculous and ineffective. However, DDoS attacks or breaches of privacy are not.
Only if it offends you. That's really the test for offensive - does it offend you?
Marking something as offensive doesn't remove it instantly or anything, so there's no harm done if you mark something as offensive and you're the only one offended.
Mark as offensive and downvote I suggest?
And also users with more power should be able to "censor" or delete the post.