Any wood that shows up in someplace like Home Depot will have been kiln dried. Unless it has gotten wet in storage or transport, it shouldn't have excessive moisture.
Pine 2x stock from your hardware store is only kiln dry to something like 16 percent. This is not acceptable for doing picture frames and is the reason the wood warps as soon as you get it home and into a dry area. You should get the cabinet grade pine if you are getting it at a hardware store.
I use walnut and cherry for frames but just about any wood will do fine as long as its not warped. I go for straight grain wood when choosing picture frame stock regardless of the variety.
Right. The 2X stock is for construction, not indoor use such as furniture or picture frames. Furniture stock and wood flooring is usually dried down to 6 to 7 %, and I'm sure the same would hold for picture frames.
The best way to dry it down further is to cut it to slightly oversize, then leave it in your house for a year, at the same temperature and humidity as the house itself (not the basement in other words).
If you buy mouldings or furniture stock, it should already be dried to the proper moisture content. Be careful of cheap mouldings though, they are often made from either MDF or finger-jointed. The MDF has no grain (but is very stable), and the joints of the finger-jointed wood show up unless painted.