I would think if it failed to converge then you would get no solution at all although I am not familiar with SPSS (I run SAS). You would get no results, not just no significant ones (indeed the p value would not show up since no solutions were arrived at). Multilevel models commonly don't converge if you have many random effects. Four is about the most you can have according to some authors and have convergence (it looks to me that you only have two a categorical dummy and the intercept although its not clear to me how many dummy variables you are using). You might want to reduce the number of random effects in your model, say start with just the intercept and rerun the results. Then if it converges add more.

Type III test tell you if a categorical variable as a whole had an impact in SAS rather than one of the dummy variables you split this into. I have not heard this applied to the test of fixed effects itself, but SPSS could use different language or tests than I am familiar with. Its possible to have a random effect be significant even if a fixed effect associated with it is not I think. I believe that some would consider reporting the random effect in that case because it shows there was significant differences in results by group, although personally I would not care about that if the variable itself failed to predict what I was interested in (that is if the fixed effects had no importance).

If you don't know multilevel regression you should read up on it before you run any models. I spent several months doing so and I am still working through it. There are several long threads on this topic on the regression and applied statistics forums which might help some. I suggest Hox, J. J. (2002). Multilevel analysis: Techniques and applications. Mahwah, NJ: Lawrence Erlbaum.