No, some people wanted to connect 'Minoans' to Kura-Araxes, for their own reasons, which is something I did not ever understand. We had a G2b labeled 'Kura-Araxes', a J1, there was an R1b of some short.
To my understanding, it is J2 that is normally associated with Kura-Araxes, or at least Hurrians, but I could be wrong.
Anyway, I know that there was an attempt to link the Etruscans and possibly the Eteocypriots to the Hurrians as well. I think people just want to bundle everybody together because it's easier to understand. I'm not saying that this is right or good though.
Now, if we assume that pre-PIE was spoken in S. Anatolia, there could have been early pre-PIE related expansions east and west but I wouldn't place any special importance on that.
Halaf culture btw, to me, always looked 'familiar' in a way. First we should understand the origins of Halaf, what changes during the Halaf-Ubaid transitional period if and when influences from the north appear etc
I do not connect 'Steppe admixture' to early / proto-Indoeuropeans. Data do not show any such connection, imo. Among the steppe-ists only Parpola has a scenario that seems plausible but my first option would be to invert it, at least.
Okay. But are you talking about Pre-Proto-Indo-European (i.e. Indo-Hittite) or are you talking about Proto-Indo-European? The former seems to have been native to the South Caucasus. The latter seems to have been a mix of the former+EEHGs. Not all Indo-European groups came from the latter though, at least the Anatolians may not have (Reich asserts that they didn't, Damgaard also asserts that they didn't). The term Proto-Indo-European really should be changed to something like "Early Modern Indo-European" or "Steppe Indo-European" and the Pre-Proto-Indo-Europeans (AKA Indo-Hittites) should be re-christened "Proto-Indo-Europeans".
The problem with this is that it could be pushed back further, which means the terms would have to change/be reapplied yet again. If Laroche and Bomhard/Fornet are correct, the Hurro-Urartians were Indo-Europeans who broke off before the Anatolians. So the Indo-Hittites+Hurro-Urartians would be the real "Proto-Indo-Europeans". The Indo-Hittites would have to be renamed something like "Hattio-Indo-Europeans".
Honestly, I think that PIE will be reapplied to the South Caucasus and the Yamnaya/Steppe derived groups (i.e. most/all surviving IE dialects) will be termed something like "Steppe-Indo-Europeans" at some point in the next few decades.