Yes, a 7 AH battery will easily supply 7 amps for 10 seconds. As for how long... well, the 7 Ampere-hour rating is usually based on a light draw, so it'll be derated a bunch if you pull 7 amps. But, it should easily run for 15 minutes and probably much longer. Read the data sheet for the battery.
Charging: basically, you feed it like you do a regular car battery, only the current must be limited to about C/5 (where C = AH capacity). So, for this battery, the current must be less than 1.4 amps. Easiest way to do this is just a resistor, sized at something like R=V/I, or 2.5V/1 A (to be safer),so about 2.5 ohms. So, try a 2.7 to 3.3 ohm resistor, or larger (within reason). Power rating of the resistor: P=E*I, 2.5V*1A=2.5W, so use at least a 5 to 10 watt rated resistor. The 2.5V comes from the assumption that the car voltage will be around 14.4 V when the alternator is charging, and that a discharged/low gelcell will be around 12V.
The diagram in this thread shows how to arrange a diode and the resistor to charge the battery and provide backup. Using Sealed Lead Acid battery with Casetronic C134