Periodicity of sum of discrete signals
First if y[n] and x[n] were periodic, then wouldn't the ratio of their periods always be rational since each period is always integer? 


No. I mean. Yes.

$x(t)$ might be e.g. $2\pi T$-periodic, and $y(t)$ could be $5T$-periodic ($T$ being the sample period). The choice of using capital $N$ for the period was a confusing one, but signals don't need to have integer sample-multiples as periods.

BUT, and that's where I'm with you, if you say $x[n]$ is periodic, it means the sampled signal $x[n]$ is periodic – which it can't be, if the original sample would have a rational-amount-of-samples period. 

And thus, you're right: if the sampled signals are periodic, they need to have a rational ratio in periods, and thus, their sum will always also be periodic.