Does old latex and oil paint (and shellac) expire?

Well, they're new enough that they won't have lead.

The latex and other water based paints may have a problem, though. Nothing can grow in the oil based products, but very bad fungus and mold can grow in latex paints. It can smell like stinky paint in the can, but when you paint it, the stench won't dry out and will never go away.

This has been very frustrating for people who have it happen, because it defies efforts to "seal over" it. Attempts to seal just don't work. They should, but they don't. Often people have to strip all the paint off, which is a stupendous amount of work.

In my experience, old cans of latex paint tend to end up with rust inside, making the paint useless. When it's not that, it's mold as Harper mentions. I would throw out everything latex based without even opening them.

Disposal practices: If it hasn't spoiled like Harper mentioned, there are some places that will recycle good latex paint. Otherwise, the can can be just opened and dried out. Then tossed in the trash. You can buy powders to speed up the process.

Oil based paints are considered hazardous waste. Check around your area, Best to find a Haz Mat disposal day and take them there.