What state should the various pins be in while the AT89Cx051 chip is inserted (or removed)? Vcc and Vpp (RST) are both at 0V, but what about the P1/P3 pins? Should they by pulled down or left floating (high)?

Whether AT89Cx051 would tolerate high state depends on internal circuitry. As long as we do not know any details, high state is strictly forbidden.

Is the XTAL1 pin (used to increment the address counter) supposed to draw current? When I set the XTAL1 pin high (controlled by a P3 pin on the 80c31), it doesn't get very high... I tried adding a 4k7 pullup, but it still only reaches 2.6V.

Are you sure you 'hit' the XTAL-input?

Good luck,
Kai