1.
If you adjust the amplifier to give 0V output when the thermocouple is just beginning to give some output...
then it makes me wonder if a thermocouple can produce negative voltage?
2.
A helpful strategy is to draw a graph of input V versus your desired output V.
3.
If you don't want negative output, then give the op amp a single polarity supply. Use a model of op amp that can work on a single supply, meaning that it can produce output all the way down to 0 V.