"Pleadings" are the written and verbal 'arguments' made in court by either side in a case. "Discovery" refers to requesting access to the accumulation of evidence by the opposite side in a case.
Yes - conviction is a judgment made by jury (or judge in a bench trial), but conviction requires a trial. A guilty plea eliminated the need for a trial.
A fact is something that can be checked and proven. A theory is a possible explanation for something. When discussing a theory, speculative words and phrases are often used such as:
may: It seems...