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Irregular watering can cause this. Are these watermelons in the ground or in containers? We recommend watering twice a day for watermelons in containers, but for ones in the ground a few times a week is all that you would need. This article will help you:
https://www.gardeningknowhow.com/edible/fruits/watermelon/watering-watermelon.htm

Blossom end rot will cause them to rot on the ends and is usually attributed to a lack of calcium and/or irregular watering. There are a few things you can try for blossom end rot treatment. Water evenly, not too much or too little. Add a low nitrogen fertilizer to the soil. he s for optimal calcium uptake. Add lime to balance the soil’s pH if it is too low, (should be between 6.0 and 6.5). Add some gypsum to the soil for additional calcium. In addition, remove affected fruit and use a calcium rich foliar spray on the plant.

A lot of that depends on the variety of watermelon you're growing. With the larger varieties, I've found it best to take all but the strongest plant for a larger melon. Leave the secondaries, they are needed to help get nutrients to the whole plant.
The number of melons on the vine with the smaller varieties really doesn't effect the size.
And taste is governed by the availability of water and other nutrients.
Melons are heavy feeds and I side dress with compost weekly for good results. Nothing tastes better than a home grown watermelon ^-^

You may not have enough bees to pollinate the blossoms. As you may know, bees have been hit by a virus seriously reducing their numbers. Try to keep all competing weeds away from the plants - the bees may be visiting their flowers instead of the watermelon. Also the use of pesticides has cut our pollinators! Watermelon have only one day to be pollinated so there is a bit of luck involved unless you hand pollinate blossoms. Do not use high nitrogen fertilizers since they produce all vines and no flowers.

Yes, as with many garden fruits and veggies, watermelon plants produce both male and female flowers, so you can enjoy fruit with only one plant. You will, however, need some pollinators (bees, butterflies, etc.) for pollination to occur unless you pllant on hand-pollinating these plants yourself.

I suggest you apply neem oil to your watermelon patch. Sounds like you have insect damage. For the lawn, I suggest you contact your local extension office (found in the gov't section of your phone book) for more specific help with your problem. Brown patches can be symptoms of various diseases, and they can help determine exactly what the problem might be and the best way to control it as well as help it to green up.