Is there a service (free or paid) that can provide the following:
1. List of most visited websites by U.S. state.
2. Keyword search in google by U.S. state.
In short, no, Nielsen Web Rankings will give you the most detailed summary of websites across the us/world/etc but it's based on a data collection augmented with statistical modelling.
Most paid analytics companies offer benchmarking tools which allow you to compare your website with others in the sector that they have data for Google analytics does this for free and is likely to have more data than anyone else.