My question is very soft but I think answerable: are the problems posed in American Mathematical Monthly, say, research type of problems? I suppose not since it is certain that these questions have solutions, but the problems can lead to more questions and then we can say it is research, is this correct?
Your first question is answered by a definite 'No', because submission of a problem must be accompanied by a solution.
See AMM, paragraph "Proposed Problems or Solutions", from where
"We invite readers to submit original problems that will appeal to readers of the Monthly. Problem proposals must be accompanied by solutions and any bibliographical information that will assist in their review. Submitted problems should not be under consideration for publication elsewhere."
is taken. This endorses your assumption.
Your second raised expectation is correct, and there are certainly instances of posed problems including solutions stretching out to new research.