How to formulate Traveling Salesman Problem (TSP) as Integer Linear Program (ILP)?
You must have entered your ILP program into the ILP solver wrong.  It's easy to verify that the ILP program is feasible, using the procedure I outlined in the comments.  In particular, from the tour $c_0 \to c_1 \to c_2 \to c_0$ we get the assignment $e_{0,1}=e_{1,2}=e_{2,0}=1$ and $e_{1,0}=e_{2,1}=e_{0,2}=0$.  Setting $u_1=0$ and $u_2=1$, we find a feasible solution: you can easily check by hand that every one of your equations is satisfied.  So, you've done something wrong in entering it into the tool.

In the future, you should do that kind of sanity check on your own before asking here.