Perkins is far behind not only the Beach Boys, whose studio influence is immense, but most of the others mentioned here. I love Carl, I think "Everybody's Trying To Be My Baby" is one of the coolest songs of its time - "They took some honey from a tree/dressed it up and called it me" - but like a lot of artists he's given more credit than he deserves for influence because he's name checked a lot by subsequent big-name stars.
1) (Let's just clarify that Carl Perkins didn't write "Everybody's Trying To Be My Baby," although IIRC he did add the line you quoted.)
2) If, as you say, the Beach Boys influence is immense, then I'll ask you the same question I asked Creep: can you please name 10 well-known artists they clearly influenced?
3) I'm not sure what you mean by "studio influence," could you clarify that?
1) Perkins contribution to the song helped define it though, but actually you're arguing against yourself, since he also didn't write Matchbox.
2) Well, to list the obvious, the Beach Boys influenced the rise of surf music, or rather surf-themed music, Jan & Dean switched styles after playing with them at an early show, and that, while short-lived (1962-1964) was still a huge national phenomenon. They were the Beatles biggest influence in their most creative period, particularly Paul McCartney, who began utilizing the melodic bass lines that Wilson made famous. Just sticking with fellow HOF'ers - R.E.M., The Ramones, The Who, Elton John, The Bee Gees, The Eagles, and interestingly enough even Metallica and Holland-Dozier-Holland have talked of how he altered the possibilities of arrangements and more complex songwriting.
3) The studio influence can be found in any artist who began constructing songs in the studio, instead of just playing the song live straight-through, Wilson built it in sections, the most obvious example is Good Vibrations. He also wasn't using traditional instrumentation all the time, weird percussive techniques, playing piano strings by hand, like a harp, theremins, etc. It led to greater freedom to experiment when his forays into this were successful. Obviously as studios progressed and more tracks became available, overdubbing and so forth artists were going to experiment, but Wilson was considered by far the most daring of those in the mid-60's, John Cale of the Velvet Underground worshipped him for this reason.
The thing about influence I think that people get wrong, not that this isn't a sign of influence, is trying to spot artists that sound exactly like another, which to me is imitation and very limiting for creative artists in general. All artists do this to a degree, especially at the start of their careers (Beach Boys did with their Chuck Berry meet Four Freshman approach) but it's the little quirks of writing and recording music where the most influence is found. Something just like the unique chord changes in a song like "Let Him Run Wild", where musicians for years have been blow away by that otherwise obscure non-charting B-side, because it doesn't "go" where it should melodically, it takes all these left turns, but still winds up working. Those are the things that turn artists heads and its more likely that you'll find similar experimentations in their work while still allowing the band to sound like themselves, whereas a direct copy has artists sound exactly like the artist they're influenced by, which generally doesn't mean a long period of relevance for the second generation artist. Both are valid ways to influence others, but most influence is more subtle in nature it seems.