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The first thing to do is to take a specimen of the fungus like stuff to a local nursery or perhaps local university horticultural extension office. Have whatever it is truly identified. They they can suggest a good fungicide that will get rid of the fungus, mold or whatever it may be. If you have no such places close by then you may want to try buying a fugicide named Immunox or one of the Ortho fungicides and apply it per the instructions on the label and see if it gets rid of it.

This article should help: https://www.gardeningknowhow.com/plant-problems/weeds/dallisgrass-weed.htm

This article should help: https://www.gardeningknowhow.com/plant-problems/weeds/hairy-bittercress-control.htm

Here is an article you may find helpful: https://www.gardeningknowhow.com/plant-problems/weeds/how-to-kill-grass-naturally-kill-unwanted-grass-in-your-yard.htm

In a perfect world, yes. While it should come up without any issues, there are no guarantees when it comes to unexptected problems. Even when you do everything right, in the end Mother Nature dictates the outcome. Hopefully, your grass will be fine and become established before the heat of summer. Just keep it well watered and cross your fingers!

You can use a post-emergent herbicide. Read the label carefully to make sure the product targets only broad leafed weeds, but not grass. This article will help you: https://www.gardeningknowhow.com/plant-problems/weeds/post-emergent-herbicides.htm
You can also seek advice on the best herbicide to use from your local Extension Service. This link will help you find one: https://www.gardeningknowhow.com/extension-search/