I have a 45 degree slope I plan to plant creeping thyme on. Lately,
the weather has not been cooperative for rototilling and planting the
slope. But my main question is, what is the best weather condition for
applying herbicide to kill the lawn grass on my slope so I will have an
easier time rototilling? Also, how long do I wait after applying
herbicide before I rototill?

A rototiller on a 45o slope? You've got your accident and disability
insurance paid up, don't you?

The best weather conditions for using herbicides are a calm day (less drift)
and the temperature conditions specified by the herbicide. All are not equal.
Most of the post emergent herbicides work best when the plants are actively
growing, so a warmish calm day a couple of days after a rain, temps between
50 and 90 is a ballpark recommendation *but you're the guy with the
herbicide* -- it's *your* responsibility to read the label and
use it properly. It's also going to tell you how long before replant.

Now personally, I wouldn't rototill. You're just bringing up more weed seeds
to give them a chance, and loosening the soil for erosion (are you sure
creeping thyme will hold the soil in your worst rains? It won't do it
where I'm from.) Instead I'd either use some glyphosate, wait for stuff
to die, and then trim the vegetation low and plant plugs between the remaining
roots, or I'd solarize and then trim and plant plugs. If I truly felt the
need to loosen the soil, I'd dig by hand. Much safer.