Part 1
I'll use the signal that I picked for the last homework to demonstrate the sampling rate idea. My signal was tan(t). If you sample this function at a rate of $ pi $, every sample will be identical, as long as it's not shifted by $ \frac{\pi}{2} $ as $ \tan(\frac{\pi}{2}+n*\pi) $ for any integer n is undefined.
However, if you sample this function with a period of anything OTHER than $ \pi $ then you get random dots all over the place.
Part 2
I picked a pretty easy function for a non-periodic one for homework 1, so I'll use it again! :) I chose $ y=x $ According to the definition, $ y(t+k*T) $ should be periodic for any k and constant T, so lets see. We get a sum that looks about like $ t + (t+1) + (t+2) + (t+3) + ... $ for T=1 and k=0,1,2,3,4...
