Oddly, no, as long as the product actually does what it is advertised to do, it is not false advertising.

Currently, it's a misappropriation of likeness issue, which is currently a state-level civil action in some but not all states. The point of the proposed law is to make stuff like this regulated by a single, federal law instead of a patchwork of state laws.

No, just as using a lookalike or (for radio ads) voice actor with a deliberately similar voice and phrasing wouldn’t, unless they actually claim it is Joe Rogan endorsing the product.

Bette Midler won a lawsuit over a soundalike in a Ford commercial. Ford first approached her to do the ad and when she turned them down, they hired an imitator.

https://en.wikipedia.org/wiki/Midler_v._Ford_Motor_Co.

Yes, but that's not on false advertising grounds, its on California state common law right of personality grounds.

The question about whether it was a false advertising violation, not whether it might be actionable under state law on some other grounds.

I guess you’d have to ask his wife