I'm away from my liquid nitrogen supply for a few days, but attached is a couple of print outs using Cory's library in DEBUG mode, hopefully it means more to you than me!
Maybe :). Unfortunately the values are good, the chip doesn't recognize any fault.
Have you found any schematics of that board? In my opinion it's a bit strange that they say that the module operates with Vcc and logic in the range 3V3 to 5V. The chip operates only at 3V3 (according to the datasheet, page 2: absolute maximum ratings for Vcc is 4.0V), so maybe that information is just wrong and you have to drive it with not more than 3.3V to get reliable results. It's possible that the module includes a voltage regulator but we don't know it. You might measure the voltage between pin 1 and 4 of the chip during operation.
On the linked page another user complained in a similar way, he also doesn't get correct temperature readings but with an R type module. I probably would try to power the module with 3V3, use a level converter for the SPI pins and see what result I get then. I don't know if you have a level converter on stock. An ADC (the MAX 31855 basically is a rather exact ADC) always needs a reference voltage to compare to. The MAX31855 is able to produce that internally but I don't know if that's exact if the supply is out the specified voltage range.