For Q1, I would have probably done the problem the same way you did it, and the work you did looks correct to me. However, as I look at the limits specified in the problem (part a) more carefully, I see there is an absolute value indicated. I find this specification very confusing, but I think they are defining 8 different box regions with those limits as follows.
1: 0.1 < x < 0.2 ; 0.1 < y < 0.2 ; 0.1 < z < 0.2 ;
2: -0.2 < x < -0.1 ; 0.1 < y < 0.2 ; 0.1 < z < 0.2 ;
3: 0.1 < x < 0.2 ; -0.2 < y < -0.1; 0.1 < z < 0.2 ;
4: -0.2 < x < -0.1 ; -0.2 < y < -0.1 ; 0.1 < z < 0.2 ;
5: 0.1 < x < 0.2 ; 0.1 < y < 0.2 ; -0.2 < z < -0.1 ;
6: -0.2 < x < -0.1 ; 0.1 < y < 0.2 ; -0.2 < z < -0.1 ;
7: 0.1 < x < 0.2 ; -0.2 < y < -0.1 ; -0.2 < z < -0.1 ;
8: -0.2 < x < -0.1 ; -0.2 < y < -0.1 ; -0.2 < z < -0.1 ;
However, the charges in these regions add to zero because some regions have positive charge and some have negative charge. The answer of 27 uC is more than a billion times smaller than your answer, so perhaps the answer should be zero, but 27 uC is due to round off error in a calculation.
Or, perhaps there is a typo in the problem. Or maybe we are both misunderstanding something. I'm not sure, but it is probably not a big deal to worry about this problem.
Q2: Integrating over the universe just means "over all space". In rectangular coordinates, the limits are -∞ to +∞ for all three dimensions. In cylindrical or spherical coordinates you need to span the entire range of the relevant variables. I'm not going to explain in detail because you mentioned you never did cylindrical or spherical coordinate integration. This needs to be studied formally, not by me making passing comments. But, to answer your question, r=sqrt(x^2+y^2+z^2) as you said.
Other query: Just because you have a negative sign at the intermediate stage in the calculation does not mean the final answer must be negative. Work it out carefully and you will see that the negative sign goes away in later steps.