My lab achieves better results than another lab on the same problem – should we collaborate?
First, it not just to say "Let's us collaborate with them!" That's not how collaborations start in the first place. To get a collaboration started you need several prerequisites: Genuine interest in the same problems (seems to be given here), similar knowledge and similar goals but also complementing expertise, being open for new ideas from the other side, and most important: Genuine interest in a collaboration in the first place.

Based on the above, I would never approach somebody (or some group) with the question "can we collaborate on this" but always start differently. The start may be different in different cases, it can, e.g., be just informal discussion with an individual without any particular goals besides getting to know each other, it could be a meeting of the workgroups with the purpose of exchanging ideas or it could be an invitation of somebody to give a talk. To see if this may lead to a collaboration is the second step…


  Should my research group collaborate with them?


I can't answer it, and probably nobody here can. If you think that a collaboration can be beneficial, than go ahead and contact them.


  Is this a proper way of collaboration?


I don't see an answer here either, since I do not see any collaboration yet. I can say that working on the same problems may be a basis for a collaboration. What would also be helpful, would be if your groups attack the same problem with different methods so that combining expertise may be helpful.


  Should they feel that our group is weaker than they are and that this is why we are trying to collaborate (a feeling of insecurity)?


I don't think that anybody will jump to this conclusion just because you asked for collaboration. I would also not jump to this conclusion based just on results for one single project (but you told that your results are even better than their results…).