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Misunderstanding Schenkerian Analysis

(Please excuse if I am incorrect with this, I am trying to learn Schenkerian analysis and there are a few things I don't quite get.)

I think the main gist of Schenkerian analysis is that all pieces can be (notated? rewritten? condensed?) into the Ursatz, or *3 blind mice*.

I get that there are several ways to elaborate on such, including 
diminution (filling in a leap between notes), 
horizontalization(turning a vertical stack of notes to a more horizontal one), 
harmonic transformation (subtly modifying a chord), and
displacement(moving one voice with respect to another)

However, I'm having trouble analyzing some compositions with just these. First of all, none of the above operations seem to produce a note higher or lower than what they started out with?
And the operations seem to spread notes out, so how do chords stay intact? And are there other ways for notes to be created besides diminution?
I'm pretty sure that counterpoint is heavily involved here, but I'm not sure where.
Are there other operations that *form* chords from a single note, or perhaps a type of diminution that includes neighbor notes? I know I'm missing something, but i'm not sure what.