By Borel-Cantelli lemma, almost-surely there is such that, for all , , hence .

It is correct that the value of on is arbitrary; they choose it to be 0. It couldn't be defined as the limit of the series since it may not converge.The proof goes on to use that last line to construct a set N of things for which the sum is infinite and then defines the limit r.v. to be 0 when things are in N, and when things are not in N.

I could also use a little explanation in understanding why this r.v. was so defined, because the subsequence constructed in the proof is supposed to converge with probability one to this random variable but I don't understand whats going on well enough to see it.

In order to be rigorous, it is important to define everywhere (as for any r.v.), even though only the values on N really matter.

This is wrong. Being "fundamental in probability" (a not so common notion, btw) is equivalent to converging in probability to some random variable.convergence with probability 1 to X is equivalent to the sequence being fundamental in probability.