How can I give power to Arduino Uno using a 12V 7.5 Ah Lead-Acid battery?
The input voltage on the DC pin is specified as 7 to 12V. Your battery may well be supplying more than that, causing the voltage regulator to overheat and shut down.


  The voltage was 12.62 volts.


That is out of spec for the Arduino Uno. When you operate out of specified voltages anything can happen, and that anything may not be very good.

You could get a buck converter to lower the voltage.

Certainly a Uno should be able to run for a long time on a 7 Ah battery (provided you are supplying voltages in range for it, preferably using a buck converter or similar to make sure that a fully-charged battery does not supply too high a voltage).