Is it possible for me to connect the solenoid to the counter, counter to transceiver on one arduino and the push button to a transceiver on another arduino? I have built the circuts, and I have a bit a a psedo-code going, but I don't even know if its possible or where I should start if it is.
Probably. But your description does not provide any details.
What Arduino(s) are you using?
What transceiver are you using? What range does it need to cover?
If you are building a wireless Arduino device, how will it be powered?
What is the purpose of the solenoid? If it is for opening a lock then a servo would probably be a lot easier to use as it can be controlled directly from the Arduino. You will need something between the Arduino and a solenoid as the Arduino I/O pins cannot provide the current needed by a solenoid.
What display are you going to use? One that is controlled with I2C will only need two Arduino I/O pins thus leaving more available for other purposes.
If you want help with code that is not working exactly as you want please post the program that represents your best attempt and tell us in detail what it actually does and what you want it to do that is different. It will make it much easier to focus on the parts you need help with rather than wasting time on things that you can do.