Can a judge give a defendant probation?

It depends on what kind of charge. The judge cannot give probation after a conviction on certain cases. You should listen to your lawyer's advice about whether to have a jury or bench trial. Generally, unless the judge is a really great and super fair jurist, I would go to the jury if there are guilt / innocence issues. Remember, judges are elected officials and generally more worried about reelection than being a fair judge - sad, but true.