Depending on the distance between the gate and your home, as well as the gauge of wiring used for the gate’s bell button, you may be able to use that cable for powering an ADC V720W found here: http://suretydiy.com/product/alarm-com-outdoor-night-vision-camera/
The ADC V720W requires only two conductors to apply power, so as long as your bell wire is of sufficient gauge and your wifi signal can reliably reach the gate, this may be a less labor intensive way to get a camera out to the gate’s location. The difficulty would be in properly mounting the camera.
A wireless 2Gig doorbell could be used as a replacement for the current gate doorbell found here: http://suretydiy.com/product/2gig-wireless-doorbell/
This would be a workaround solution to function with your Alarm.com account.
Wireless and wired intercom systems exist that may better fit your needs here, but I know of none that are compatible with Alarm.com. It’s not something we currently sell on our DIY store but standalone devices that our full-service team have used include the iVision intercom, a pdf of which can be found here: http://www.optexamerica.com/sites/default/files/resource-uploads/iVision_Lit_New.pdf
For the gate lock, however, if an electric lock is used (perhaps taking advantage of the existing bell cable instead of a camera) it should be possible to wire it to a zwave switch relay and a proper power supply inside your home in order to have remote control as well as schedule capabilities.
With only a two conductor bell wire to take advantage of, the choices for easy installation are a bit limited, but there are options.