A {} in math mode inserts an (empty) ordinary atom, which can influence the spacing. Let's see how.
$ - \alpha {} - \alpha \underbrace{\alpha} \cdot
\omega \underbrace{\alpha} {} \cdot \omega $
Note that the \quad spaces don't influence the automatic math spacings, so we can disregard them. This becomes the following sequence of atoms:
Bin Ord Ord Bin Ord Op Bin Ord Op Ord Bin Ord
and two "Bin" atoms are transformed into "Ord" because they're not between two atoms which they can operate on. Indeed the \underbrace macro creates an Op atom:
\def\underbrace#1{\mathop{\vtop{\m@th\ialign{##\crcr
$\hfil\displaystyle{#1}\hfil$\crcr
\noalign{\kern3\p@\nointerlineskip}%
\upbracefill\crcr\noalign{\kern3\p@}}}}\limits}
Thus the spacings are those dictated by the rules:
{Bin} 0 Ord 0 Ord 2 Bin 2 Ord 1 Op 1 {Bin} 0 Ord 1 Op 1 Ord 2 Bin 2 Ord
where {Bin} means a Bin atom transformed into Ord, 0 is no space, 1 is thin space and 2 is medium space.
Let's see your final examples:
${\underbrace{\alpha}}\cdot \omega$
This is "Ord Bin Ord", because the braces around the Op atom transform it into an Ord one.
Ord 2 Bin 2 Ord
$\underbrace{\alpha} {} \cdot \omega$
This is "Op Ord Bin Ord", so we get a thin space after the Op, a medium space after the Ord and a medium space after the Bin:
Op 1 Ord 2 Bin 2 Ord"
$\underbrace{\alpha} \cdot \omega$
This is a bit harder. We have "Op Bin Ord", but the Bin becomes Ord for spacing purposes, because it's similar to $\log+2$. So we get a thin space after the underbraced alpha and no space between \cdot and \omega.
Op 1 {Bin} 0 Ord
