12/2 connected to 8/3 wire with extra wires taped off. Is this legal?
As a tap to a second building with no sub panel NO It would not be legal. 
The term tap usually means connecting a smaller conductor to a larger feeder and there are some cases where a tap is legal but not here with the info provided.

Maybe it’s legal if this is spliced not tapped if the breaker feeding the 8-3 is 20 amp it probably is ok 

(they may have run larger wire to reduce the voltage drop) If the breaker is any larger than 20 amp it is not legal any size breaker larger than 20 feeding this would need a sub panel at the shop/shed sized for 20 amp max because of the 12 awg.

Updated for OP update.  First if “normal wire” like Romex from the splice it is not legal from the splice. prior to the splice a 15 amp breaker the ampacity is fine and the breaker could be larger, voltage drop is not a hard code issue it is a recommendation in the form of a fine print note not enforceable. 
My concern is the type of cable. Back to the home if UF it has 4 conductors but 1 is capped with a wire nut.

If it were me I would get some 8-3 with ground UF and a proper inground splice kit.  Extend the feeder to the shed and terminate in a minimum of a 50 amp main breaker sub panel. Drive a new ground rod and connect it to the sub.
Upgrade the 15 amp single pole to a 40 amp double pole. Used the capped wire and the wire on the 15 amp breaker in the main panel to feed your sub panel. 
You asked if the “taped” connection was legal, I would guess it is not because it would need a listed in ground splice kit, these are fairly simple some have 4 butt connectors with an insulator that shrink tubing with an adhesive sealant (very inexpensive. Others are higher in cost where the splices are staggered with butt splices and each connection is sealed with shrink tubing with the adhesive sealant , the last type is the most expensive but the splices are made a plastic structure goes around the cable and epoxy is mixed and filled into the plastic and let dry (most expensive)  I use all 3 types and in your case if the box is not always under water go with the least expensive and it will be fine. 

So upsize the splice, add your new panel and new ground rod and you will have a nice service, 8 awg wire 125’ is 2.88% voltage drop at 40 amps so you don’t even have to worry about voltage drop.