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There is no weed killer that will kill monkey grass but not turf grass. You can spot treat the monkey grass with Round-up and a spray shield to kill it that way or hand dig it out, but you will have bare spots in your lawn for a bit that you will need to watch in order to keep other weeds from moving into those areas until they fill in.

Grass is very difficult to grow under any tree, but there are a few things you can do. First, have the tree's branches thinned out. This will improve the light. Next, have the soil tested and treat it for whatever the test recommends. Third, make sure that the area is getting adequate water. Trees drink a huge amount of water and in order for the grass to grow or remain healthy, you will need supplemental water under the tree.

This article should help with getting rid of this weed: https://www.gardeningknowhow.com/plant-problems/weeds/eliminate-dollar-weed.htm

It could be voles, moles or even snakes, all make dirt mounds that are just a few inches high. If the mounds are smaller than a few inches high, it may be ants or possibly termites. Likely, it is from moles that are looking for food, often digging and burrowing in the lawn. They are nocturnal animals, so it is likely you would not see them. Here is more information on controlling moles: https://www.gardeningknowhow.com/plant-problems/pests/mole-control-home-remedies-for-removing-moles-from-your-yard.htm

If you think ants may be behind the dirt piles, this article should help: https://www.gardeningknowhow.com/plant-problems/pests/insects/get-rid-of-ants.htm

Since I don't know if you are asking for identification or eradication of the plant, I suggest taking a fresh sample to your local agricultural extension office for both identification and recommendations.