Weird behaviour from S1J-13-F Diode
I would not expect 1.1V at only ~11uA, more like 0.5V (simulation says 444mV). 

If it has 6V across it when forward biased, it's either blown up or will be within milliseconds. You might be able to see evidence on the surface like a hole. Moderate overloads typically make a diode fail shorted, but Lipo battery has sufficient current capability to cause a small explosion most likely. 

Or perhaps the parts are counterfeits and are bereft of internal die. 

Presumably it also measures 6V when the erstwhile (or never-was) diode is entirely removed, which indicates your meter has rather low input resistance (maybe a really cheap one?).