Is there any good recommended software/websites to design your own layout?
...
2D CAD: QCad
Circuit board design: KiCad.
Then copy and paste from existing keyboard designs that are open source.
bpiphany wrote a thread a year or two ago about how he designed his custom keyboards. There is a lot more detailed info in that one, if the thread has not been deleted for some reason.
You should be able to find much more info in the "Making stuff together" forum.
Is it better to have the switches mounted on a PCB rather than wiring if you're going to be using a plate
anyway?
If you use direct wiring and intend to be able to swap keycaps some day, then they can't just be snapped to a plate or you risk pulling the entire switch from the plate when you had intended to pull just a keycap.
You must either have a PCB or glue the switch bottoms to the plate, decide to keep the same keycaps indefinitely, or be very very careful when you swap keycaps.
Some keycaps and switch combinations are more tight than others.
4. Which stabilizers do you recommend for MX Black plate-mounted switches?
5. Some keyboards utilize different switches for the spacebar. Is it recommended to use a different one? If yes, which one?
Those are questions of preference.
For the Space Bar, it is usually a heavier switch that otherwise has the same characteristics. MX Brown -> MX Clear, MX Black -> MX Dark Grey, MX Blue -> MX Green, MX Clear -> MX Grey.