as a rule, yes
Yes - consecutive serial numbers make them more valuable, when sold.
59-63
Serial numbers do not designate calibers. Look on the brrel to ID the caliber.
Usually caliber and/or serial numbers.
No. Serial numbers aer NOT UNIQUE. They are intended to be unique to a given make and model- but a gun can be rebarreled to a different caliber.
Impossible to answer without a serial number, maker and detailed description.
The purpose of a serial number is for identification, regardless of the merchandise.
The value of a five dollar bill is five dollars. It's not clear what you mean by "consecutive serial numbers." If you mean that the bill's serial number looks like "12345678" or something, then a collector might be willing to pay a small premium for it as a curiosity. If you mean that you have two or more five dollar bills where the serial numbers are consecutive, they're worth five dollars each, period.
No way for us to tell from a serial number alone. Serial numbers are NOT unique to any one firearm.
Probably late 1880's
Impossible to answer.
1966 I have the same one serial numbers and all