Citation or inclusion of name in conference paper
That's a delicate question. My own rule is to offer a co-authorship to anybody with whom I discussed the problem unless I use only the results they have already published and took the official credit for. However I don't insist and if they decline, I just give them extended thanks in the paper. So, I would definitely contact the (leader of) the previous team and offer to include all the names in the author line.

As to ow to approach your professor, it is not entirely unlikely that he is reading this very thread at the same time as you already, so once you decided to go public with it, just show him the replies you get here and ask what he thinks and what his arguments are. If he is a good man, he'll be straight with you and hold no grudges. If not, well, then you are in trouble already...