As seen in the picture, how would I have the quotes on the right be mirrored (or whatever else would need to take place)?
You have two buttons for this in the transformations palette: 
There are several quote glyphs. Make sure each has the right direction. If you are not sure, use the UnicodeChecker tool to see how other fonts have it. You can active it by clicking the arrow in the top right of the grey info box.
miss type problem? maybe…
'
single-left 「Option」＋「[」
’
single-right 「Shift」＋「Option」＋「[」
“
double-left 「Option」＋「@」
”
double-right「Shift」＋「Option」＋「@」