The TC1262 is rated to 6.0V input voltage. Using a 9V battery is a Pretty Bad Idea. It seems you're really lucky that your regulator is still working.
Drop-out voltage is indeed very low, making it suitable for use with a 3.7V LiPo battery (actual output would be 3.3-4.2V depending on the charge level).
To boot: CH_PD, RESET, GPIO2, GPIO0 high, GPIO15 low.
GPIO2 will be pulled high through the internal pull-up, probably GPIO0 as well, I'm not sure if you have to actively connect those two pins to the desired levels. Adding a pull-up resistor won't hurt anyway.
To get the chip into program mode: boot with GPIO0 pulled low, the rest unchanged.