Honestly with how weird the Bardiel fight is I feel that the ending was less 'The Dummy Plug is very functional' and more 'the Dummy Plug forced a berserk state so Unit-01 won'.
Eh, that seems to be a rather extravagant additional assumption, IMO. As it is, "the DP worked" is perfectly workable as explanation and requires no extra assumption. And yes, the DP is entirely autonomous, and not a remote control for the EVAs. It literally is a mind-copy of Rei
(that is the purpose of her mind upload, which is done only once and that in preparation of the DP - I think the common fan conception that Rei gets regular mind uploads to give to her clone if need be is wrong. Since Kyoko and Yui clearly recognize and hence remember their children we can assume memory follows soul). So we basically have a more mindless version of Rei fighting there. But that was in fact enough to defeat Bardiel.
Whether it would have been enough to beat Arael - well, a piloted EVA couldn't do that either. It did require the Spear of Longinus. In fact, since there is no soul for Arael to mindrape, a DP EVA might even fare better here. Armisael... well, Rei did use the self-destruct system. A DP based off her might do the same. Though of course, it does seem a more mindless and less inhibited version of her mind, so who knows. And as for Kaworu - all you need is Rei's AT Field to cancel out his. Then even a gun would do the same job as the EVA hand in canon. Kaworu is an angel where you a) can't use an EVA to defeat the angel but b) don't need one, either. It all comes down to Rei.
The total unconnection between Rei and Kyoko wouldn't matter, since Shinji is shown in episode 06 to be able to sync to Unit-02,
Only in dual-synch with Asuka. I think the facts that Asuka pilots the EVA with her mother's soul inside, Shinji pilots the EVA with his mother's soul inside (and can pilot the EVA with his mother's clone's soul) and Rei pilots the EVA with, uh, her inside (and can pilot the EVA with her genedonor's soul inside) does mean that something. Plus of course that class 2A is the class of potential pilots and nobody there seems to have a living mother. It does seem that in order to synchronize, you do need to have an EVA with the pilot's mother's soul inside. If cross-synching were that easy, most likely EVA-02 would have been part of the cross-synch test.
Given that, I conclude that the only reason that Shinji showed a synch-rate with EVA-02 was because he was in there with
Asuka. Plus, the cross-synch test was literally testing if Rei (and hence her DP) can synch with EVA-00 and EVA-01. If EVA-02 had truly been meant for a working DP, they would have put her in EVA-02 and tested that as well; but that didn't happen.