You may or may not have a paper thin washer. If you did have one, it may or may not be on the drawing. Those shims are added when dimensions don't add up correctly due to normal dimensional tolerances. They are not always needed. They are often, but not always, added at the discretion of the assembler at time of assembly.
Spring pins can stick out of one side of a hole & still be OK as long as both shear lines where the parts meet have full diameter pin engagement. If that explanation is not clear, let me know & I'll find another way to describe it. If the pin was sticking out so far that only one of the two shear lines was covered, then that is a problem.
A pin that does not want to come out may indicate a problem. When those pins are overstressed, they sometimes yield partially. In that case, the pin is no longer straight. In extreme cases, a shoulder forms at the shear line that will block the pin from coming out in either direction. If that has happened, the holes may have wallowed & may need to be redrilled to the next larger pin size. If the shafts are hardened, then the holes are probably fine as-is, & would be very difficult to drill with normal tools anyway.
I would let the stuck pin sit in penetrating oil for a few hours, then have at it with the punch again. A dead blow ball peen would be my hammer of choice if one were available. Be sure that the underside of the shaft is well blocked with wood against a solid surface, like a concrete floor or an anvil or an iron keel
. Use a short pin punch that is very close to the diameter of the pin, but just a bit smaller.
If that doesn't work, then your best next move is to find a machine shop with a Bridgeport & a small carbide end mill to cut out the bad pin. Failing that, you could try to make a hole through the pin with a drill bit, but that is likely to do more harm than good. An end mill in a Bridgeport will go in a straight line. A drill bit in a drill press will wander. The pin is going to be a little hard. You may need to heat it up to around 400 or 600f to soften it enough for a drill bit to get through it. Heating
the pin enough without overheating
the shaft can be a bit of a trick.
Once you have a hole through the center that is about 60% of the pin diameter, you should be able to punch out the remaining part of the pin, even if it is damaged. Boring out the center should relieve enough pressure to let the pin move.
Hopefully, the pin will come out with the punch after it sits in oil for a while. Drilling the pin out, without making a mess of the job, is not easy.
Dan's suggestion of leaving the old pin in there may be a good idea if the pin does not want to come out with a hammer & punch.