throttle plate is under flywheel, and that moves by design. Rough running and shaking sounds like ignition issues, offhand. Need a compression test with a screw-in style gauge. You will also need an adjustable spark tester. If compression is good, and you want to work on the motor, change out the impeller as the motor is new to you. A lot depends on how comfortable you are (or want to be) in working on it. Get a manual.
If you know the history of the motor, that can potentially help with decisions on what to do first. Maybe the recent owner replaced ignition coils, for example. Best to just figure on rebuilding the carburetor with an appropriate kit -- new float is good, but not absolutely necessary. You may need new ignition coils, with new points and condensers -- depends a little on what the history is. You may need the driver coil that is under the flywheel.
Post a model number for the motor.