Electrical – Will an arduino output pin set low ground a pin on a DFPlayer module

I think this can be done with just a couple of diodes and a resistor.

Pressing the button with power off enables PWR_ENABLE turning on the micro (or power supply or whatever). The micro asserts POWER_HOLD to keep the power on. The micro then monitors PWR_DOWN_REQUEST to see when the user has pressed the button again. If VBATT is above the level that can be fed into an input of the micro, a voltage divider can be added.

According to the document you provided, the datasette port is looking for a pure digital signal with varying duty cycle (0.75 for H, 0.25 for L).

As long as the Arduino pin can drive sufficient current (it should be able to) and is operating at 5V, a direct connection will work. You may want to investigate using a TTL buffer between the Arduino and the C64 (the buffer would be powered from the +5 supply of the datasette port, and the ground would be common to both the C64 and Arduino).

As for the SENSE, it would be easier to use a digital output to drive a small-signal MOSFET (like a 2N7002) - a logic high turns the MOSFET on, which pulls the SENSE pin (connected to the drain) to ground (connected to the source) without the Arduino having to sink any current at all.

The MOTOR pin could also be used to drive a MOSFET gate. The drain would be pulled up to the Arduino supply voltage with a weak pullup (10k or so), the source connected to ground. The drain would also go to a digital logic pin. When MOTOR is high, the logic input is low, and vice versa, and the Arduino sees a clean logic signal.

For example...

Note using two NAND gates as a buffer of sorts. (Can you tell I used to scrounge for parts?)

TTL is fairly robust. I don't think there's much chance of damaging anything.