As we saw in the films, it took until *after* the battle of Hoth for the name "Skywalker" to reach the Emperor (or at least, for him to get around to telling Vader) which by all accounts is several years after Yavin.
Based on Vader's reaction, he must have assumed (thanks to Palpatine's goading) that Padme died right there on Mustafar and "the baby" along with her.
One assumes there may be other Skywalkers out there, descended from siblings or cousins of Shimi, so the name may not be massively unusual.
No. It's established that Vader already knew about Luke prior to his conversation with the Emperor in at least two different ways. First of all, there's the opening crawl, as I already cited.
Secondly, when the fragment of the probe droid's report plays to Vader, he says that he is sure "Skywalker" is with the Rebels, dovetailing with and confirming that what the opening crawl says is literally true: Vader is searching for Luke prior to the Imperial fleet's arrival at Hoth.
I said at least two ways, and there really is a third. Vader chases after Falcon, evidently believing Luke to be aboard her, at least until it arrives at Bespin. It must have been Leia's presence that he sensed, and he merely confused it with Luke's.