Studio 60 On the Sunset Strip/WMG

Tim Batale is a real person
The singer claimed that narcotics made her confuse real and imaginary conversations. But even if we accept that and apply it to Matt, she never mentioned creating entirely imaginary people and thinking they were real. Therefore, there really was a Tim Batale working for Studio 60. It's just that the weirder conversations with him were Matt's imagination, and Tim was too stoned to show up for the group photo.

NBS shut down shortly after the series finale
Because the writers started the FCC plotline and never resolved it.

NBS is what the DuMont Network would have become if it had survived to the present day
Because it has too long a history to be an upstart - it was around during blacklist season - and because it competes with networks that also exist in Real Life. NBC exists, Fox exists...
 * Corollary: the world of Studio 60 still has studio-owned movie theaters.
 * I'm pretty sure that Studio 60 mentions being in the old DuMont theater at least once in the series.

In the show's world, the US never invaded Iraq.
As you'd expect from Sorkin, there was lots of discussion of the War on Terror and US politics in general on the show. So it's odd that neither Iraq or any location within Iraq was ever mentioned. The simple explanation is that in the show's world, the US invaded Afghanistan but not Iraq. We already know it's an alternate history, with the NBS network and the history of the studio. When Jack rages about "fighting two wars" he's referring to Afghanistan and somewhere other than Iraq that the US is fighting.
 * Ya had me until the last sentence... if there is another war going on, and they haven't made any references to that one, how is the situation any different?

The show is set in the same world as the Watchmen movie
"NBS" is the network in the Watchmen movie's viral advertising. After the end of Watchmen, history continued on a course that resulted in a world almost identical to our own.

Matt is a time traveller.
Matt is the only one who remembers Tim Batale (who was clearly erased by the Crack) and looks exactly the same age in the flashbacks as he does in the 'now'.

The sketches aren't as bad as we think.
Or at least not as bad as the ones before Matt Albie showed up.

The series begins with the guy in charge of Studio 60 having a nervous breakdown on air and lambasting all of television because the one sketch he liked got cut. And after Matt takes over writing, people say the Show Within a Show is "the best its been in years."

Maybe Studio 60 was just so abysmal before that the mediocre sketches we see are gold compared to what Studio 60 aired in the years we're not shown.