Glueing PVC pipe is not really a glueing process, it is a chemical welding process whereby both materials are softened by the glue and melted together in the same manner as a thermal weld would attach metal.

The primer removes the glazed surface of the PVC in the area of the chemical weld to allow the PVC glue to penetrate into the pipe & fitting wall to insure the correct penetration.

Failure to use primer results in substandard faulty welds that may or may not withstand the pressure and vibrations of normal use.

I see the primer is for a chemical welding process. Should the primer be applied a little wide than the glue is done? For example, the primer is applied 4 inches wide while the glue is done for 2 inches wide on the pipe. In addition, should the primer and glue be applied on both of the pipe and joint or only on the pipe? Can you answer two questions above? Thank you.

Primer is applied to BOTH joining surfaces, i.e., inside the fitting socket, and outside the pipe, to the socket depth.
No, the primer needen't be "a little wide", but no big deal if it is. DON'T let the primer puddle in the socket.
On small diameter tubing, I don't apply glue to the socket, just the pipe (under 1&1/2") unless I can't twist the joint together. Don't puddle glue in the socket.

Can I use finger to quickly apply the primer and glue on the pipe and elbow joint or I must use something or a special tool to do it, by the way?

"Can I use finger to quickly apply the primer and glue on the pipe and elbow joint or I must use something or a special tool to do it, by the way?"

For lines as small as you have the applicator is inside the can of primer and cement attached to the lid.

PVC/CPVC cleaner, primer and glue ALL contain solvents that at the very LEAST will de-fat your skin. Read the labels and follow the safety instructions.
Avoid skin contact.

If your cans don't have a built in applicator, you can use an artist's or small paintbrush preferably with natural/nylon bristles and a wooden handle.

Because it is a solvent, the primer will also clean the surfaces of both items.