Why ~would~ Conan use Jay's old set and band?
The Tonight Show isn't meant to be some ready-made and packaged show in a box that's passed from host to host (like your example, The Price is Right, is). Every host has their own distinct style and mark that they put on the show; it truly does become a different show altogether. (tangent) Again, this isn't the case with The Price is Right. That show IS Bob Barker's even if Drew Carey is hosting. It's not a show that changes and evolves; it's still the same show it was in the 70s, right down to the stylized decor and logo, so this comparison is faulty. And it's not a show that HAS to change with the times anyway like a late night talk show has to; the only things that MUST change are the items and price tags. (/end tangent) The Tonight Show is even passed to an entirely different production company. At the end of the day, the emphasis isn't truly on "The Tonight Show" (although it really has become an iconic franchise); it's really all about the "with ______ ______".
Jay didn't keep Johnny Carson's band, so Conan wouldn't be expected to keep Jay's. You have to think about the chemistry between the host and the band... I really can't imagine Conan bantering back and forth with Kevin Eubanks. And their music just would not mesh at all with Conan. It's actually kinda funny just to think about it. And let's not forget the fact that Jay will have his new show in September. He probably doesn't want Kevin and the gang going anywhere.
In regards to the studio, a BIG reason they keep The Tonight Show in CA is because The Late Show with David Letterman is based in New York. There'd be some major tug-of-war games with booking guests if both shows filmed on the same coast. So it's not just an issue of tradition only. And Conan isn't using Leno's set for the same reasons he's not using Leno's band. Besides, The Tonight Show is now being filmed in Hollywood at Universal Studios; Jay's Tonight Show studio was in Burbank. I really wish Conan COULD'VE stayed in NY though. It's more along the lines of his personality and humor, and I love the feel of that city SO MUCH more than LA.
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ETA: I think I already answered that question if you read it thoroughly. And the explanation for the theme and closing music is the same as the explanation for the band. :/
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ETA2: Wow this is turning into an essay haha :)
Why the production companies change-
Biggest reason-- the production companies are either fully or partially owned by the hosts. Jay owns Big Dog Productions, Conan owns Conaco (just for the record, Letterman owns Worldwide Pants, which obviously produces The Late Show). And also with that the entire STAFF changes, from the writers to the directors to the interns.
About the hosts having a "package deal" re: theme songs-
Never said that Conan COULDN'T have had a new theme and closing song. Just said that he wouldn't use ~Leno's~ songs. Max Weinberg and TPTB could've very well composed entirely new songs if they had decided to; instead they just embellished the old one.
And I couldn't tell you if The Jay Leno Show in September will use Jay's Tonight Show theme (they might compose a new one too), but Kevin Eubanks has all but officially confirmed that he'll be staying with Jay and the band is fully expected to follow suit. They studio will even be on the same lot as Jay's Tonight Show, so it's not like they'd have to make a huge move across the country like Conan's band. :)
And yep, The Jay Leno Show in September is expected to have many similarities to his Tonight Show (Jaywalking, Headlines, monologue, guest interviews). It's supposed to have more focus on current events and politics though.
Why didn't the show names just stay the same, while the schedule moved up an hour instead-
Leno had already resigned his position, then came back after it was all in writing and changed his mind. Conan had already signed a contract in 2004 with NBC which granted him The Tonight Show this year, and everyone was fine and dandy with that, including Jay. He had stated he'd be done with television by then. It was only four years after the contract was signed (last year) when Jay started having second thoughts and didn't want to leave. So they gave him a new primetime show so he wouldn't defect to another network. Letting him keep The Tonight Show would be a breach of contract.
About guest-booking wars b/w Late Night with Conan and Late Show with Letterman-
Late Night and Late Show aren't competitors, they're at different time slots. The Tonight Show and The Late Show ~are~ competitors, and they attract the same caliber of guests to boot. Shows like Late Night, Kimmel's show, and Ferguson's show typically have guests on that aren't as "A list", for lack of a better term. :P
Anyway, sorry if this is long-winded. Trying to be as concise as possible while still fully answering all of your questions, but that's kinda hard this late. :)