Can a pin be Neutral, not high or low?
What you are talking about in this instance is "Open Drain".  Normally, as you rightly note, with outputs you have either HIGH or LOW, which effectively connects the output pin to +5V or GND respectively.

Instead you are asking for Open Drain.  In this arrangement a pin is either "active", in which case it connects it to GND, or "inactive", in which case it is disconnected from (almost) anything.

Most 8-bit Arduino boards don't have Open Drain capability on their IO pins (some more advanced 32-bit boards do, though), but it is easy enough to emulate it in software.

You can emulate it using "Tri-State" logic.  That is logic where, as well as HIGH and LOW you have "High Impedance", or "HI-Z".  HI-Z is pretty much the same as "inactive" Open Drain. On the Arduino HI-Z is the same as INPUT.

So you can emulate it by:


To activate a pin set it to OUTPUT and LOW
To deactivate a pin set it to INPUT


For example, to "press" one of your buttons for 100ms you could:

pinMode(4, OUTPUT);
digitalWrite(4, LOW);
delay(100);
pinMode(4, INPUT);