Maybe should add the need and importance of using a jack stand and not relying on just a jack alone! And no need for a puller. Big punch or socket extension and hammer will get the shaft out of the hub. Also the axle nut needs to be torqued with a torque wrench. (major cause of repeat failures). Clean up the knuckle where the hub goes and use anti seize before putting it back together.
As far as the vibration when hitting brakes most likely its your front rotors are warped. If you feel a vibration at certain speeds but goes away at other speeds than i would suspect tire balance or a bad tire.