I am trying to make a power-on reset circuit that can also be manually reset. The reset is active-low. When the capacitor is charging, I'm assuming 5v will be present on pin 10. Would this damage the IC?
No, that will not work. The output will drive the capacitor voltage high or low relatively quickly (maybe a millisecond) and the resistor will do nothing of value. You'd need a gate or to short the capacitor with a switch + small value resistor.
Ignoring the top inverter, this kind of Rube Goldberg R-C reset only works some of the time and has hazards that may not be immediately obvious- the capacitor can discharge into protection networks of the inverter under some conditions (such as a heavy load or short on Vcc), destroying the chip. That can be patched by adding a resistor and a diode but that still won't make it a reliable reset circuit.
I suggest using a proper reset chip that is guaranteed to work and give an adequate length reset pulse (eg. 200ms) under all conditions including brief interruptions of power, extremely slow rise or fall of power, etc.
To specify that you need to know the minimum voltage your circuit will operate correctly at (where can reset be released) and the maximum voltage at which you can be assured that no ill effects will result from reset not being asserted. And the minimum reset pulse width. If there is insufficient overlap to allow the use of a commercial device with its tolerances you may have to rethink clock speeds or device specifications etc. to allow operation at lower voltage.