In the most general sense, a common law system is adversarial; a contest between two competing views. Objective truth is less important than who wins. A civil law system is inquisitorial; the point is to find out what the truth is.
Civil law jurisdictions don’t really allow judges to make law. Judges rule solely based on the applicable legislation*. Judges in common law jurisdictions consider not only legislation, but the earlier rulings of equal and higher courts.
Most countries combine civil and common law, including the “classic” common law system, England and Wales.
The prosecutor is perfectly free to request an acquittal in common law systems, too. In fact, the prosecutor can effectively grant an acquittal by dropping one or more of the charges. A prosecutor’s job is to “do justice”, not to obtain convictions.
Of course, as you correctly point out, the trial would not need to continue in that case.
Well, that depends. The prosecution (or plaintiff) bears the burden of proof, but if that burden is met, the defense bears the burden of production- tipping the balance back in their own favor.
*I gather many civil law jurisdictions now allow for judges to create law, to a degree.
A common law marriage is one in which the parties hold themselves out as a married couple despite not having solemnized their union (ie., taken vows). The term is a complete misnomer- the practice was largely disfavored at common law- and I have no idea where it came from.