Can a dismissed misdemeanor cause me to miss out on this job?
Not at all,
most employers don`t care about misdemeanor crimes.
If they only ask about felonies,
would they even check for a misdemeanor?
Misdemeanors include things like parking tickets,
most employers don`t care enough to check.
They do have to pay to do background checks,
if you say no to a felony,
then they won`t go further.
Will a dismissed misdemeanor automatically show up on a background check?
As long as it was expunged.
Even if they ask you something as broad as "have you ever been convicted of a crime (including misdemeanors)" you should still say no!!
Its been expunged,
they shouldn`t be able to find it.