Can I replace Arduino with Raspberry Pi and still use the same circuit board?
Your diagram shows power to the motor and solenoid coming from the Arduino Vin.  The Raspberry Pi has only a 5V supply.  Your motor and solenoid will probably need a separate power supply when used with the Pi.
The MOSFETs you have (looks like 30N06L) will work with 5V, but I have seen posts from other folks suggesting that the 30N06L doesn't (quite) work with 3.3VGS.  It may work OK(ish) or it might not.  It depends to some extent on your load.  Some 30N06L also just work a little better at lower VGS than others.

You're going to have to change at least the power source for the motor and solenoid, and you may need to use a different MOSFET.