Dear Amanda and Lynn,

 I can see you have a real problem with the grass where the city did their projects. One of the first concerns is that the disturbed soil, when replaced, might not have all been topsoil. Sometimes not all the topsoil gets back on top when filling in projects. I might su ggest doing a soil test before you begin the process of renovating your lawn. Here is information on how to go about getting a soil test. Most county extension offices have soil test kits available for a nominal fee.

Soil Testing for Horticultural Needs | Ohioline

After preparing the soil, selecting the new grass seed appropriate for the location is important. Here is information on selecting the proper seed for the area. 

Lawn Grass Cultivar Selection | Ohioline

Addressing the weed issue, I expect the best approach for weed trees is hand removal, if you do not wish to use herbicides. The removal of other weeds may be selective. A dandelion is a weed to some and an early pollination plant to others.  I  did find some information on organic lawn care which may prove helpful.

Natural Organic Lawn Care | Ohioline

There are some videos from Ohio State which may also prove helpful and I have included the link to them here:

Renovate your lawn; How to, step by step | Fairfield

I'm sure  that an early start on your lawn renovation will give you greater success. It has been an interesting spring for Ohio gardeners. Watch the weather for watering needs in your new lawn 

Wishing you the best,