Will potash burn my lawn if I apply it directly?
You are fine applying it directly as long as you follow the instructions on the package. Apply it in cool, dry weather to lower the risk of burning even further.
My lawn care service recommended that I put down peat moss on the bare spots of my lawn and this should encourage the grass to spread and fill in the bare spots. Do I need to do any prep work before laying down the moss?
The peat adds organic matter to the soil, and increases water holding capabilities. This article will tell you more: https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/peat-moss-information.htm
You might dig up the soil a bit (1/2 - 1") on the bare spots, then put down your moss, put a little soil on top of it, then water will, the purpose being to keep the peat from blowing away.
If I spread weed 'n feed fertilizer on patches in my lawn where I have spread grass seed, will the "weed" portion of the fertilizer keep my grass seed from germinating? If so, how long after the seeds germinate can I safely use my fertilizer?
You are correct in that weed & feed might prevent seed germination. It is recommended to wait until your new grass is well established and you have mowed it a minimum of 4 times (usually 6 to 8 weeks) before applying a weed and feed fertilizer.
I have a 15 X 23 area of shade area in my backyard. What type of grass should I use?
Here is an article on shade tolerant grass that will help you:
https://www.gardeningknowhow.com/lawn-care/lgen/shade-tolerant-grass.htm
What can you do with sods of grass dug from the formation of a new border? I am digging a new border and will have the grass left.
You could compost the sod.
Here are some links to help you.
https://www.gardeningknowhow.com/composting/ingredients/grass-clipping-composting.htm
https://www.gardeningknowhow.com/composting/basics/lasagna-sod-composting.htm
My gardener cuts the grass the day after it rains and the ground is still wet. I believe it should not be done. Am I wrong? Thank you.
If the blades of grass are bent over from the weight of the water then it is best to wait until the lawn is dryer.
Wet grass does not cut as cleanly and can clump up in the mower more easily.
Have laid new sod in home in Myrtle Beach, SC and would like to know how often and length of time watering is required for the first two weeks.
Here are two good links that will help you with water and care requirements.
https://www.gardeningknowhow.com/lawn-care/lgen/installing-sod-instructions.htm
https://www.gardeningknowhow.com/lawn-care/lgen/so-you-want-your-grass-to-grow.htm