Invasive? No.

Container plant? I've never tried that. But if lovage is happy, it can get to be quite a sizable plant in a couple year's time. I would suggest planting it in the ground for best results since it has large taproots.

In my experience, it prefers a morning sun or partial shade location. It seems to bake in full sun. Since you live south of me, I would definitely suggest a cooler or shadier location.

FataMorgana

DL, I've got two lovage plants growing in my front bed. It is on the north side of the house. It gets partial sun, plenty of water, and they seem to be very happy for the most part. Also, a plant the voles didn't eat!!!

I am finding lovage to be very invasive.

I snap off the flowering stalks to prevent self seeding. Give it room because it can become a pretty large plant. I never used it for cooking but the stems are kind of neat because they are hollow and can be used as a drinking straw.