If I pay to get a traffic to my website, will CJ and other affiliate programs still honor conversions made by paid traffic?
Paid traffic is fine. However, if your using something like adwords and bidding on keywords/creating ad text there are often certain restrictions that you need to read through - trademarked keywords are off limits etc...
Most of the time it's fine. But each merchant might have certain restrictions like not bidding on trademarks and certain terms, no facebook traffic, no adware traffic, etc.
Most sites will encourage it. As long as you do not violate specific TOS and can make money from it, then go for it. That's how big business works, they buy what they need and establish an ROI model.