I just changed the spark plug and filter. Took off the drive cover to check the belt. I do not know if the belt should be loose until you bring the control level toward the handle.
Generally the mower shutting off soon after starting would be a fueling issue. I would recommend taking the bowl off the carburetor and cleaning it. Also you might want to spray some carb cleaner right down the throat of the carb. The vibration could be caused by an unbalance blade, a bent blade, the damaged blade adapter, or a bent crankshaft. On most machines the belt is tight no matter what's going on with the transmission. You didn't include your model number so I can't look at your model and say for sure though.
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