2005 Honda Civic running hot after replacing the radiator, heater core, and thermostat?

Things I've noticed.

It running hot is directly tied to me turning the heater and AC on. Usually turning the heater on would cool the engine down, but it makes the temperature spike.

I replaced the heater core because it was leaking and I smelled antifreeze when the AC was on.

The radiator is brand new.

I've straight wired both fans to the battery.

I think you may have air in the system. You should check for that first.

I've also noticed that water seems to be coming out of the reservoir cap. Any ideas?

Needs burped. Bleed air, sometimes can be a pain.

Besides air in the system, it's possible that the thermostat was installed upside down - they fit only one way to be effective + the fans are meant to turn on when they are needed, NOT all the time… That will not help with the cooling!