There are various ways of removing a lawn....just pick the one that appeals to you and provides results in a timely manner. In that small a space, I'd just use a square edged shovel and skim off the lawn and its roots by scooting the shovel blade under the surface.

You can plant the ajuga at any time you can work the soil, although now - midsummer - would be at the bottom of my list. Fall would be a better time or in early spring. Ajuga grows perfectly well here in full sun but then this far north, my sun is not as intense or as hot as it might be in Nashville :-) Morning sun and afternoon shade or light, all-day (filtered shade) is ideal and with enough sun to encourage good blooming. And it dislikes very dry conditions - evenly moist soil is best.

'Chocolate Chip' is a very dwarf and slow to spread cultivar. And develops the best foliage color with a reasonable amount of direct sun.

Thanks so much @gardengal48. Question for everyone. What other varieties of ajuga should I consider?

I wouldn't consider any other, especially in full sun. 'Chocolate Chip' is a good one, but really full sun in 6b might be a little too much for any Ajuga.

I would think very carefully about introducing ajuga to your property. It is often the decision most regrettedby gardeners - up there with trumpet vine, wisteria...

Hmmmmm.........I would not put ajuga under the same category as trumpet vine or wisteria! While it does spread - that is the entire point of a groundcover - it is not at all as aggressive as many other plants and is easily removed where not wanted. And it will take full sun in my climate as long as the soil remains consistently moist, although the cultivars with dark foliage - Chocolate Chip, Black Scallop, Bronze Beauty, etc. - seem better adapted to sunnier conditions

Will this choke out daffodils? Using it as a weed suppressant