Direct contradiction can be seen as confrontational, or just emphatic. It's often more polite to phrase a disagreement in a way that shows some respect for the original thought, but diverges from it.
In your first example, "No. It's not warm." gives the first speaker no credit for a valid opinion, though I think your alternate response, "Really? It's cold." may actually be more aggressive. "Really?" can mean "Is that really true?" when the speaker has just learned something surprising. But "Really?" when rejecting something as false sounds more like "Can you really think that?" or "Can you really expect me to believe that?" It sounds pretty harsh. I'd propose a third option, "I find it rather chilly." This response, and similar phrasings, express the same thought, that it isn't warm, but by personalizing the disagreement leaves room for both opinions to be valid.
In your second example, I think the difference is emphasis. Both are telling the first speaker that their plan is a bad one, but the first response, "No. Don't use A and B to get C. No." suggests that the plan is a terrible, horrible idea and should not be considered. The final "no" seems unnecessary to me, but does add emphasis. If the first speaker was proud of their plan, then a particularly sound rejection of it might be hurtful. The second response, "Using A and B to get C isn't the best method." suggests that the plan might be okay, but there's a better one. In this case, the circumstances should determine which is better. If the plan really is that bad, a strong rejection might save a lot of hassle down the road, where someone might decide to stick with their original plan in the face of the less emphatic response.
In your third example, I see much less difference, though the first one is possibly clearer. It makes it immediately clear that the second speaker is objecting to the plan. In the second example, someone could actually be confused about the meaning of "Do that first." if they aren't sure that "that" is the same "that" from the first sentence. Depending on the context, it might not be.