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Creeping Monkey Grass generally does not need much fertilizer, but a light fertilizing will help monkey grass to spread.
I would use a 10 10 10 fertilizer in early spring or fall, before new growth begins.
1 pound per 100 square feet is recommended.
Water in the fertilizer to help it into the soil.

Here is a link about the care of Monkey Grass.

https://www.gardeningknowhow.com/garden-how-to/soil-fertilizers/blood-meal-fertilizer.htm

Monkey grass thrives in full sun and tolerates heat and drought. This article may be helpful in caring for the plant: https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/monkey-grass-a-spectacular-groundcover-for-the-lawn.htm

Offhand, no. However, Arabicus mondo grass (mondo grass and moley grass often confused) forms pink blossoms during summer but it has black foliage. With a little digging, I did manage to find a liriope variety called 'Samantha' that has green foliage with pink flower spikes.

Monkey grass generally requires very little in the way of fertilizer, if any at all. These are extremely hardy plants. In fact, some varieties can become aggressive if not situated in a contained area or given enough room to spread. Here is an article that you may find helpful: https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/monkey-grass-a-spectacular-groundcover-for-the-lawn.htm

Well it sounds like you have some creeping liriope (L. spicata) mixed in with your clumping type. This kind is quite aggressive and better left to fill in large empty spaces of the landscape. You can spot treat this invading monkey grass with Round-up or hand dig it out.

The following article tells you when to plant monkey grass:

https://www.gardeningknowhow.com/ornamental/groundcover/monkey-grass/transplant-monkey-grass.htm

Per this article, you should "wait until there is no chance of frost and it should be safe to transplant through midsummer. After transplanting monkey grass, it will need time to establish itself to survive the cold weather and after midsummer, it may no be able to do this."