What is the thinnest metallic sheet commercially available and what is the thinnest sheet of metal producable in lab?

Broadly, one can obtain a thin metal sheet by processing it—rolling or beating it, for example—from thicker stock or by depositing it atom by atom (the latter achieved in evaporation, sputtering, or electrodeposition, for instance).

It's possible to beat gold leaves to thinness to the point of transparency. According to the Encyclopedia Britannica:

The process of goldbeating is as follows: The gold...is...cast into an ingot, and flattened, by rolling between a pair of powerful smooth steel rollers, into a ribbon of 12 in. wide and 10 ft. in length to the oz. After being flattened it is annealed and cut into pieces of about 62 grs. each, or about 75 per oz., and placed between the leaves of a cutch...

The cutch is beaten on for about 20 minutes with a 17-lb hammer, which rebounds by the elasticity of the skin, and saves the labour of lifting, by which the gold is spread to the size of the cutch; each leaf is then taken out, and cut into four pieces, and put between the skins of a shoder...The shoder requires about two hours' beating upon with a 9-lb hammer.

At the second hour, when the gold is about the 150,000th part of an inch in thickness, it for the first time permits the transmission of the rays of light. Pure gold, or gold but slightly alloyed, transmits green rays...

The reported thickness is on the order of 100 nm. Faraday estimated a similar typical thickness of 1/282,000 in. See also Mallet's 1903 report "On the structure of gold-leaf, and the absorption spectrum of gold", which also notes thicknesses of ~100 nm.

These films (and modern commercially available films) are at the lower limit of what can be handled—near wisps that crumple under the slightest air current.

Regarding vapor deposition (from the bottom up), there's no lower limit to the thickness that can be obtained. I've evaporated gold films with increasing thickness from single clusters of atoms to a cohesive film several atoms thick. Such films are deposited on a smooth substrate and would have no physical integrity as a freestanding film in air. (Nevertheless, one could always deposit them on a soluble substrate—polished NaCl, say—and then dissolve that substrate from underneath to obtain a freefloating film.)