1. Instead of the ill-located JST plug, can I connect the battery's red wire to the TTGO's "VBAT" pin, and expect it to work the same (powering the board, or recharging when the board is powered by a wall plug)?
The JST connector is unfortunately the only option that retains full functionality. You can connect your battery (or other external power source) to the +5v pin and it will work, but the downsides are:
1) It bypasses the polyfuse overcurrent protection that the JST circuit includes
2) It bypasses the charger IC. Also I suspect that plugging the board into a USB might apply VBUS (minus a diode drop) to the +5v pin which wouldn't be a great idea if there is a Lipo connected. Definitely check this out.
The "BAT" pin is a switched power output for supplying external circuits, as is the 3V3 pin. Both of these pins are enabled by the presence of VBUS or a high level on the ESP GPIO 14 pin.
2. Generally speaking, are some pins hard-wired for a given use, eg. 17/TX + 18/RX? IOW, even if I change the settings in the source code, only specific fins can perform specific tasks?
Most of the pins can be configured for at least two functions, some as many as five. 17 and 18 can be configured for GPIO, UART, DAC or ADC by your code. Worth downloading the ESP32S datasheet and use it in conjunction with the TTGO board schematic on Github.
https://www.espressif.com/sites/default/files/documentation/esp32-s2_datasheet_en.pdf3. What are "VBUS" and "VDD3V3"?
VBUS is the 5v supply that comes from a USB charger or computer when plugged into the USB connector.
VDD3V3 is the output of a 3.3v regulator that is powered from either the battery or VBUS when the USB is plugged in. The VDD3V3 pins have 3.3v on them all the time that the board is powered but as mentioned above the 3V3 pin is switchable output which derived from VDD3V3.