Gap showing between baseboard and shorter-pile carpet

There needs to be a 1/4" gap next to all walls to allow the flooring room to expand.

This gap should be covered with trim of some sort. The problem with baseboard alone, is that most baseboard is 3/8" thick, leaving only an 1/8" of an inch for the wood to contract without exposing the edge. This also assumes that the flooring is laid with tremendous precision, which it is likely not.

So your options are:

1. Find thicker baseboards. Door and window casing comes in 3/4" thickness and can be used.
2. Use quarter round or shoe molding to thicken the baseboard.
3. Use door stop in the same way you would use shoe molding.
4. Undercut the drywall so the flooring can expand underneath and use regular baseboard (thanks Matthew)

I had the 3rd option done in my basement, and it turned out pretty nicely.

I would also recommend the caulk. It makes everything look nicer.

Shoe molding may have to be removed in the future to work on the floor or the wall. That's fairly easy with nails. It's fairly difficult with glue, unless you use something like hide glue that's specifically intended to allow disassembly ... and the techniques used for that would be hard to apply in situ to parts of a house.

Generally, don't glue anything that may need to be serviced, unless you're willing to risk having to destroy it to remove it.