Parties often try to settle cases, before filing litigation and during litigation... even after a judgment But nothing can force a person to settle. Everyone is entitled to a day in court. Once a case is filed its up to a third party (judge or jury) to evaluate the "evidence which proves without a doubt that they would win." either by summary judgment or trial. There are some delays in terms of when a motion can be heard with this "evidence" but a summary judgment motion can determine the case without trial.

It all depends on the potential defendant's attitude toward the controversy and upon the attorney he or she hires. Some lawyers want to defend to the max. Others will suggest to the client to settle before the defendant incurs substantial attorney's fees and costs. I think the best approach is to get a feel for the situation by having an attorney do so on your behalf so that you're not doing anything to harm your case.

Sometimes yes, sometimes no. Not all litigants are interested in settling. And many who are will insist upon terms that the other parties won't accept.

And it is extremely rare for one party to be able to "prove[] without a doubt that they would win" if the case went to trial. It is much more common for parties to mistakenly believe they have such proof.

You got a lot of long-winded answers. The simple answer is that no legal dispute ends on its own until one of three things happens: The plaintiff/claimant/aggrieved party voluntarily gives up; the defendant/disputant/objecting party voluntarily admits responsibility; or a court issues a judgment and options for appeal are exhausted. The only other way to end a dispute is compromise.