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You are going to want to roll your lawn in the spring when your grass is just coming out of dormancy and the roots are in active growth. You should mow your lawn in the spring when it gets to be about 3 to 4 inches tall. These two criteria are not likely to happen simultaneously so I recommend that the two processes (rolling and mowing) be done separately with rolling being the first priority of spring.

For more information on rolling out grass lawns, please visit the following link:
https://www.gardeningknowhow.com/lawn-care/lgen/how-to-roll-out-a-grass-lawn.htm

The use of gypsum on residential lawns is not recommended due to the cost and side effect reasons outlined in this article:
https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/using-gypsum-in-garden.htm

It is instead recommended to incorporate compost into your soil instead as it provides nutrition, water holding capacity and improved soil structure - and is much cheaper.

However, if you do want to pursue applying gypsum, you should follow the directions on your bag of gypsum. If there are no directions printed on the bag, the home depot has a chart on their website that will tell you how much gypsum to broadcast based on square footage:

http://www.homedepot.com/c/gypsum_lime_fertilizer_HT_BG_OD

This article will give you different things you can try:
https://www.gardeningknowhow.com/lawn-care/lgen/controlling-algae-in-grass.htm

It probably is lawn slime mold. For more information on lawn slime mold and a course of treatment, please visit the following link:
https://www.gardeningknowhow.com/lawn-care/lgen/black-substance-on-lawns.htm

This article will help you tackle the brown spots in your lawn:

https://www.gardeningknowhow.com/lawn-care/lgen/brown-lawn-fixes.htm