The upcoming Supergirl TV series is developed by Greg Berlanti, who also developed both Arrow and The Flash. The three shows also share many other crew members.
Will Supergirl be set in the same universe as The Flash and Arrow?
We now know from the 4 show crossover that Supergirl takes place on (from the perspective of Arrow/Flash/Legends being on Earth Prime) Earth-38. So barring a worlds collide event transporting supergirl to a parallel universe, her show does NOT take place on the same Earth as the other shows.
We don't know. It is being produced by the same people who created and run Arrow, Flash, and soon Legends of Tomorrow, which is why there's a real possibility of a Supergirl crossover. However, over the past months, I haven't seen any concrete statements on whether there will be a crossover. This article says it's part of Greg Berlanti's (one of the producers) deal:
Fazekas: Marc, CBS is doing Supergirl. What is the possibility of a crossover?
Guggenheim: It's possible.
THR: I actually heard it's in Greg's deal, that Supergirl can cross over with Flash and/or Arrow.
But other articles say otherwise. Nina Tassler, Chairman of CBS Entertainment said:
That's a different network so I think we're going to keep Supergirl to ourself right now.
And then later on, new Chairman of CBS Entertainment Glenn Geller said:
“I have to be really careful what I say here,” Geller said on the Television Critics Association press tour. "Watch and wait and see what happens,” he concluded.
So, it's a mystery whether or not Supergirl can and will crossover with the other shows.
Yes, as of Feb 3, 2016, the shows are part of the same fictional universe. (Note that the Arrowverse itself already includes a DC-comics-like incarnation of a multi-verse; it's highly likely that Kara Kent and Barry Allen will be from different internally-defined Earths, but part of the same externally-defined fictional reality.)
We have known for a while that this was a possibility, as the shows are produced by the same group of people, and CBS partially owns the CW. Originally the network execs were less enthusiastic to the idea, but when Supergirl's full-season pickup was announced, it was accompanied by rumors that a crossover might happen.
It has now been confirmed:
CBS announced today that Grant Gustin’s Flash will be crossing over into an episode of Supergirl later this year—a development that’s sure to please fans of both series. While details of what the crossover will entail are yet to be revealed, the episode is set to air on Monday, March 28th. src
Assuming there are no off-weeks, March 28th would be S01E20 of Supergirl, but no other details about that episode are known; the announcement also doesn't mention of the crossover will be reciprocal (e.g. there's no information that Kara Danvers will appear on the CW Arrowverse shows.) However, the description of the crossover as:
Grant Gustin for making the time to come visit
strongly implies that this one, at least, will be a one-way crossover.
Unless they open up another Verse, along with Earth 2, then I can not see how. Superman has been active for years in the Supergirl universe and would you think that would have to have been referenced by now in Arrow (especially as the whole meta-human phenomena has only started occurring and is news to everyone that "abilities" exist)
In "Legends" there is reference to Men of Steel and Dark Knights, so potentially open in future episodes to have Supes come into Earth 1