When schools do background checks they are primarily looking for employers and employment dates. It is not an issue to be hired through a contracting agency, but I think it could not hurt to mention somewhere in your application the name of the contracting agency so that the adcoms dont have to make any connections on their own.
Regarding your misdemeanor I think all of the following are true:
- likelihood of this coming out in a background check: probably low
- likelihood of this influencing your application should you disclose: also probably low
- likelihood of schools dinging you after finding this on your background check, not previously disclosed: still not very high, but probably higher than the others
It's the last possibility I would want to mitigate against.