Mint is typically propagated, as you correctly say, by runners (or stolons). Stolons are longish horizontal parts of the stem that can either grow along the surface or underground and connect two vertical stems. Here's a picture of a mint stolon from this site
(I'll replace this with pictures from my plant when I get the time).
It's the stolons that have the ability to put out both shoots (upward stems) and roots from the nodes. The upright stems do not have this ability and so if you only have mint stems from the grocery store, they're probably never going to grow roots.
The reason that you saw some of them remain fresh is because the xylem, which is the plant tissue responsible for upward water transport, might've still been alive and hence would've been transporting water to the leaves via capillary action. However, it cannot sustain itself this way, and without a corresponding upward transport from the roots, the capillary pull cannot balance the water loss from transpiration in the leaves and eventually they'll dry. Slowly, but surely.
I've not tried to get a stolon to root in water, however, I wouldn't be surprised if it did, although it's not really a common way of growing it. To propagate it effectively, you need a good section with a bit of root (not necessary though), part of the stolon (a must) and some shoots with leaves (to nourish the plant when it's establishing itself). Although highly uncommon, some varieties are propagated by seed.
Regarding its invasiveness, you can contain mint by growing it in a pot. That way, eventually it will get root bound, which then determines its maximum size. If you plant it on the ground, you better be making lots of mojitos ;)