I have a very clean, spacious (almost 500sqft) studio on the UES for $1850/Mo. It's possible to find, you just have to sift through a couple hundred apt listings of glorified closets to find something decent.

Find a good broker with exclusive listings with small-time landlords who value stable long term tenants over getting away with crazy rent increases and high turnover. I'll recommend Tosh Marks though his inventory is limited due to the exclusivity so the timing may not be right.

Biggest factor in my experience in avoiding rats/mice/roaches/etc. is to make sure you're in a building that has trash bins outside. My bro lived in East village for a while and they just piled their garbage underneath the staircase on the first floor. And no shit there were bugs and other critters.