We do not need to connect every pin of the Pi Pico.
We need to connect any GPIO pins we want to use, of course, but we do do not need to connect any unused GPIO pins. Of course, not all GPIO pins are equal; some do PWM, or ADC, or I2C or SPI or UART.
We do need to connect at least one of the GND pins to Ground. Opinions vary, but some folks suggest connecting all of them.
If you want to power the Pico via a pin, you’ll need to put +5V on VSYS.
If the Pico is getting power from the USB port, then you’ll get 5v on the VBUS pin, which can be used to power other things that want 5v. In any case, you can use 3V3(OUT) to power other things that want 3.3v.
Set 3V3_EN to low for a hardware shutdown / restart, of the Pico and anything plugged in to 3V3(OUT). Set RUN to low for a software reset of just the Pico itself.
ADC_REF and AGND are power & ground references for ADC pins.
See this page for a ton more useful info.