No, you should not expect an internal pull-up or pull-down "resistor" to behave like an external resistor. The internal device may actually be a very weak MOSFET or an implanted silicon resistor. The effective resistance is likely to vary with applied voltage. I would also expect a large temperature coefficient, and of course a very large variation ...
The 2x 10K parallel resistors on each I2C bus line is the same as having a 5K pull-up on each.
The 5K resistances are pull-ups to 1.8 V. So the I2C drivers will need to be able to sink 360 uA, which IC pin drivers for I2C chips will.
Having separate pull-ups is good if your two boards are ever disconnected from each other. The separate resistors will pull ...
You should be using a N-channel MOSFET.
Previous answers suggest using a BJT, but BJT will have about 0.6V voltage drop, and the microcontroller will probably read that as a logical 1.
The MOSFET will have almost 0V drop (assuming the current is very small) thus removing that issue.
simulate this circuit – Schematic created using CircuitLab
DP_0_PIN13 is pulled down by an 1M resistor. To effectively pull it up you need to raise it to at least 2V. The pull up and pull down resistors form a voltage divider. So anything less than 0.65M should do it.
I would go with a 470k resistor just to be sure.
The value of pull up resistor for an open drain is picked to support two functions:
Prevent too much current from flowing through the drain when the output transistor is turned on.
To provide enough voltage/current to the load when the output transistor is turned off.
You are right that there is a voltage drop across the resistor, BUT if designed ...
You should be using an NPN transistor if you are expecting it to work the same as your relay version in your schematic.
You should check how much current is being drawn through the relay contacts when they are closed and the voltage across them when open to ensure you choose a transistor that can take the voltage and current.
Probably around 10k would be sufficient. It is only a pullup for the reset signal, so not super critical. The esp01 board might already have a resistor on reset, in which case you don’t need another.
P.S from what i googled, the esp-01 has no pull-up, so 10k should do it.
Note: your battery is drawn the wrong way up.