My first inclination is that your keypad and remote have been removed from the Logic board memory.
If you have the Opener's instruction manual, then go to the programming section for how to re-program your keypad and remote.
If you do not have your manual then you can go to the Liftmaster website and download / view the manual for your particular unit. You will need the model number for your unit. That can be found near the light fixture on the unit (if you have two light fixtures, then look for a data sticker that indicates Model number and Serial number. You may have to remove or lift the light cover in order to find it.)
This is the link to the Liftmaster website: http://www.liftmaster.com/consumerweb/service/manuals/instructionmanualslanding.htm
To answer your second question, I will need to know what model of gate opener you have, and also need to know if there is a separate Receiver box unit or not. Most Gate operators will have the receiver unit built into them. Some however do not, and an external one will be wired to the gate motor unit, and possibly mounted to a wall near the gate motor.
question: do you use the same remote for the gate that you use for the Garage door opener or do you have a separate remote for each? If you do, I need the model number on the remote for the gate operator. That will tell me what kind of tx/rx you have and possibly where to locate the learn button.