I do it in early spring and/or fall mainly b/c less strenuous for me. But truly, when area around plant looks bare (just dirt) it's time to mulch.

Polechka, we mulch in the fall, after the rainy season is done. Mulch wears down more rapidly with rain, so we wait till the rain slows down markedly before we put a nice, thick coat on that will last a long time (usually October or November in central FL). Fall is a good time to prepare your plants for winter cold, and mulching over roots is part of preparing them, so fall is also good in that aspect. We don't own azaleas, but I imagine, like most blooming shrubs, the best time to prune them is right after they bloom. If you prune them now, you risk removing a lot of the buds/flowers that will form on the ends. I would wait till late winter/early spring to prune liriope. I don't really think it's necessary to prune them at all, but if you want a very nice, neat look (if winter browns the ends), late winter or very early spring may be the best time. I don't quite remember when the flowers come up, but they come from the base of the plants, so I don't think you'd disturb the flowers if you decide to cut the grassy ends then. Congrats on the new house! Wishing you the best.