You dont need to do convolution at all. This is a trick question.

For the first question, the inpulse response h(t) is a delta function shifted by time 2 units, so the output will be the shifted version of input only. Thats it, no need to do convolution integration.

For, second question, use the concept of centre limit theorem. You should know that, if you convolve, two rectangular pulse, you will get a triangular wave. So. z(t) is a rectangular pulse rect(t). You just need to find its width.

All the best 

AE