It's too soon to say if your child has "difficulties"...........most 2 year olds are pains in the bu**s....................sweet and adorable but it's definitely the "ME! ME! ME!" stage. She probably just needs firm consequences for her actions...........at this age they crave it............and of course don't forget to PRAISE her for all her little accomplishments.
And unless you have a firm plan.........a safe comfortable home, back up sitters, and the finances are in order...............I say there's no shame in staying with your parents...........that's what we're here for................when things get rough...........move in with mom and dad!
The "mom" may be a deadbeat..................but she's still the mom and unless she's abusive should have access to your child..................to avoid disappointment, just don't mention whether the mom would be coming over or not.
If your truly worried about the daughters mental state.............contact the local Community Social Services agency(or whatever it's called in your area) and ask for an assessment..............this could probably done thru your family doc too, if he agrees there may be issues
As far as housing.............if your near the poverty line..............you may be eligible for Coop housing, especially being a single parent and depending on the list and how needy your circumstances are..................but like I said before, it nice to want to move out and be "independent" but how realistic are you being?................daycare is extremely expensive..................and babysitters don't fall out of the sky