1- Do SMD components work the same as through-hole components? Should I use the same type of copper plates and everything?
SMD components work same as TH components, but, there is some difference. We use SMD components to reduce the area of them. Also, power of an SMD resistor, breakdown voltage of an SMD capacitor etc. can be changeable, you must reach them from datasheets or something else. Some THT components can be hard to find in SMD type. For example, high capacitance electrolytic capacitors.
2- For the PIC microcontroller, do I need to add a crystal to the circuit or does it have an internal crystal?
Some of the microcontrollers may have an internal oscillator, but an oscillator can be connected externally. The datasheet of the microcontroller used should be examined. When using a microcontroller with an internal oscillator with an external oscillator, programming changes may be necessary.
3- Do SMD components have any space between them and the plate for the traces to go in between (like through-hole ones)?
Pads are used in SMDs, while pads with vias are used in TH so that soldering can be done on the PCB.
4- Where do I connect the 5 V input on the PIC? I am using a 12V adaptor as power source and I'm decreasing the voltage with 7805.
After reducing the voltage to 5V with your regulator, you should connect the output of regulator to the microcontroller's VCC pin.