Common Ground between Real Analysis and Measure Theory

1. No. They are really a kind of Boolean algebra (see also field of sets). The terminology is deeply unfortunate but also deeply entrenched.

Rather than answer your other questions (they are in some sense just not the kinds of questions one asks in measure theory) let me just make some general comments. Real analysis is in some sense the study of metric spaces. Any metric space gives rise to a topological space, and any topological space gives rise to a measurable space with the same underlying set whose $\sigma$-algebra is the Borel $\sigma$-algebra generated by the open sets. Now one can ask for measures on this $\sigma$-algebra. A fundamental such example is the Lebesgue measure on $\mathbb{R}^n$, which is a very powerful and flexible way to integrate functions on $\mathbb{R}^n$ generalizing the Riemann integral.

Measure theory allows rigorous constructions of a very important class of metric spaces, namely the $L^p$-spaces. The techniques you're currently learning in real analysis will be important for understanding these spaces, which are studied in functional analysis.