I am sorry for this delayed reply to your gardening question. The spring rush has brought a deluge of questions and we are working hard to catch up!
Removing surface roots is very bad for the tree and will result in it being more likely to blow over in a storm. I suggest instead that you bring in just enough soil to cover the roots in areas where they are exposed. This will make it easier to grow grass and mow without the roots in the way.
There are several good grass choices for your area and each has its pro's and con's. The following web site explains your choices well:
Thanks for the question. Best wishes for a wonderful gardening season. Please stop in again soon!
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