0. if the pattern continues, is this a repeating decimal? why or why not?

First we observe pattern here. The pattern is one  number "1" and one number "2" than two numbers "1" and two numbers "2" and so on. Every next array of numbers 1 and 2 will have one extra member.

Because of this, 0.121122111222... isn't repeating decimal. Repeating decimal has some number of decimals that keep repeating. For example if we had:
0.121122  121122  121122 ... this would be repeating decimal because 121122 segment keeps repeating.  

In your task you can't find a segment that keeps repeating.