We don't know the origin of populus, it may be related to an Etruscan word or it may be related to an Indo-European root for "many." The Etruscan explanation I don't think makes sense, especially any connection with the city of Populonia, since the original name of the city was almost certainly Fufluna. The variant of populus written popolus perhaps suggests an Etruscan influence, but it need not be an Etruscan word any more than the variation of Claudius (a Sabine name) written Clodius need be. Populus is almost certainly related to plebs, with which it can be sometimes interchangeable, and that word is certainly derived from a Proto-Indo-European root "ple-", "fill" (whence Latin pleo, to fill). Greek has the same word πίμπλημι, "to fill" and its derived adjective πλήρης "full" (and the related adjective πλέος). Compare the related Greek terms πλῆ θος, "multitude" and πολύς, "many."

Not exactly. It has been suggested that Pupluna (Populonia) comes from the same root as Latin populus/Umbrian puplum. If correct, that would make them cognate.

The existence of the root in both Latin and Umbrian may look like it suggests a Proto-Italic root, but it doesn't have any cognates in other IE languages, so it's certainly possible that it's non-IE in origin. Margaret Watmough, Studies in the Etruscan Loanwords in Latin (Florence, 1997), pp. 91-102, suggests an Etruscan root *puple, based on the city name of Pupluna.

But that doesn't mean the Latin/Umbrian word comes from the city name: rather, both the city name and the Latin/Umbrian words would all come from the same Etruscan root. That is, supposing the argument is correct.

That root would also be the origin of Fufluns: the root of populus shows up in religious titles in Latin, too (the Poplifugia festival) and in Umbrian (puprikes, epithet of the god Puemune), so it looks reasonable to see Dionysus getting an Etruscan epithet in the same way, and that epithet superceding his proper name.

But I should emphasize that the link drawn by Watmough is pretty speculative.