The extra grill cloth looks like oxblood, which would not be original. It would be very unusual for regular blackface grill cloth to have darkened to that maroon a color. But I suppose the photo may be picking up some red from something else. Put it up against a white background. But the grill cloth on there looks fine. OK, it's not original. It will never be perfectly original again, that's the way it is.
The stuff on your finger does look like normal oxidation of the chassis. I've seen many, many old Fenders from the blackface/silverface era with this type of oxidation.
The speakers look original - 1966 Oxfords (465 speaker code, 636 = 36th week of 1966).
Serial # looks right for 66 via Greg Gagliano's serial number dating page -
http://www.ggjaguar.com/fendamp5.htm
I suggest you look at the transformer codes - typical Schumacher transformer codes will look like 606-YWW, where Y will be the last digit of the year of production, WW will be the 2-digit week of production.
Definitely get it to a
good tech who is very experienced with old Fenders. Avoid general electronics guys with a soldering iron but little to no experience with old tube amps. Better to wait for the right tech than let a hack screw it up. Let them evaluate things, including the electrolytic filter caps. Originals generally need to be changed, but if they were replaced with good ones at some point, they may be OK. This stuff can be tested, a good tech will know how to do that.
I strongly recommend that you do NOT let them change all the signal-path caps/resistors/etc., just for the sake of replacing them. If the electrolytics are good, I would evaluate the other components by how they sound, and replace only the ones that are needed due to leakage, out-of-spec, etc. And if replacing, I suggest replacing with caps/resistors as close to the originals as possible. For my money, putting modern-design caps to try to cut the noise floor down affects the sound in an adverse way. Just my opinion, but I've seen too many people do this and be unhappy with the result. I like these amps to be as close to the sound of when they were new as possible.
These are great amps, IMO.