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This article may help: https://www.gardeningknowhow.com/lawn-care/lgen/eliminate-mushrooms-in-your-lawn.htm

It may be quack grass or perhaps Bermuda grass. This will help with quack grass: https://www.gardeningknowhow.com/plant-problems/weeds/killing-quack-grass.htm

While some would say fall planting is desirable, I would suggest waiting until spring to plant this warm-season grass. Generally, sprigs and sod will not produce roots unless the soil temperature exceeds 55 degrees F for several weeks.

I honestly could not say for sure as costs vary greatly by location. I would recommended contacting a few different landscaping companies and asking for a quote. Having multiple companies in the area quote it for you will give you an idea of what the average rate in your area is and whether this is reasonable.

Yes. You can do this. Typically, people will take plugs of grass from one area to another. A plug is really just a small piece of the lawn cut out.

I suspect that the 'not connecting properly' is being caused by a fungus or a pest. This article goes over several possibilities and how to deal with each: http://www.ces.ncsu.edu/depts/pp/notes/oldnotes/tg1.htm