From their latest official publication:
Note that the Arabic name is simply a translation of בית המקדש, beit hamikdash, or House of the Temple.
So it did exist!
As for "Sham", as JE Dyer blogs:
“Sham” is largely coterminous with what the West calls the Levant*, but it has meanings for Islam that are both historical and eschatological. Al-Baghdadi, a doctoral scholar of Islam and one-time cleric from Tikrit, Iraq, grew up steeped in those meanings, as thoroughly as it is possible for anyone to be.
Of special significance among those meanings, as I outlined in June (link above), is the legacy of Abraham, which – in a form altered from the text of Genesis – is foundational for Islam, and which unfolded across exactly the territory Al-Baghdadi has been busy conquering over the last two years.
...The role of al-Sham
With that in mind, see what Dabiq has to say about the territory of al-Sham (p. 9; emphasis added):
Sham is the land of malahim [that is, battle; the implication, in particular, is that it is the land of eschatologically significant battles – J.E.]…
Allah’s Messenger…linked this blessed land with many of the events related to al-Masih, al-Mahdi, and the Dajjal.** …
Allah’s Messenger…said, “I saw as if a pillar of the Book was taken from underneath my pillow, so I looked and it was a shining light extending towards Shām. Verily faith, at the time of tribulations, is in Shām.”
Allah’s Messenger…said, “Shām is the land of congregation and dispersal [meaning resurrection].”
Shaykh Hamūd at-Tuwayjirī…in commenting on some of the narrations about the tribulations and battles in Shām, said, “In these narrations is evidence that the bulk of at-Tā’ifatul-Mansūrah (the victorious group) will be in Shām near the end of times, because the Khilafah [caliph] will be there. They will continue to be there clearly upon the truth until Allah sends the pleasant breeze and it takes the soul of every person who has faith in his heart…”
Sham is thus clearly framed as the geographic heart of the Islamic eschaton. There is a strong sense that Al-Baghdadi sees a divine hand in the current opportunity – offered by the turmoil in Syria and Iraq – to reestablish an Islamic caliphate with its center there.
“Sham” is not Syria (nor is it “Greater Syria,” as outlined in the post-World War II political concept). It’s bigger than Syria and includes a sliver of southern Turkey, parts of modern-day Jordan and Iraq, and all of Lebanon, Israel, and the Palestinian-Arab territories. Sham was a way of referring to a geographic region rather than a defined political entity, and as such, it existed inside the various caliphates that succeeded Mohammed. (In this, the regional designation “Sham” was actually similar to the way the Greeks and Romans referred to the same region, with variations on what the Romans came to call Syria Palaestina.) It was during the Rashidun, or “Rightly-guided,” caliphates in the 7th century that Muslims adopted the expression Bilad al-Sham, to refer to what was called Canaan in the time of Moses, and what Westerners now call the Levant.
I guess we're smack in the middle of it all.