Are facial 'cleansing' products 'effective' on human skin, and if so, what is their mechanism of action?
Do facial cleansing products actually do what they claim?

This dermatologist-tested cleansing gel formula lathers into a bubbly foam removing surface dirt, oil and makeup, while energizing exfoliators sweep away dead skin cells.

So essentially, they are claiming that it is a foaming soap.
Sodium Laureth Sulfate is a detergent, Lauryl glucoside is a surfactant, so it is a good bet that the product is effective as soap. There are a bunch of other compounds that modify how it forms films, so it probably also foams.
If so, what is their mechanism of action (i.e., how do the ingredients work)?
Similar to any other soap.  They make non-water soluble things on the surface of you skin (such as oil, make up, dead skin cells, etc) more water soluble so that it dissolves when you wash your face.
Some of the other you linked also have moisturizers, so they probably put another oil back in the place of the ones you removed to keep the skin from getting too dry.

I am unsure if any of these terms are reflective of a real problem, or if the products can even even help said issues.

Or do they work better than normal soap of comparable strength?  Probably not much, although having them labeled as use for your face is nice in that you have a reasonable idea that it won't be too harsh or irritating, whereas if you tried dish soap on your face that might not go so well.