As for the TARDIS, in EVERY new series regeneration she's thrown some sort of hissy fit within moments of the Doctor doing the lightshow, and then crashed somewhere. Only twice has she also suffered catastrophic damage, but at no point has the Doctor shaken off the last few sparks of a regeneration, flipped a switch and had the TARDIS continue on her merry way.
I got the impression that the fender-bender the TARDIS had in "Born Again"/"The Christmas Invasion" was more the result of the Doctor's impaired driving, and the TARDIS's reaction to his residual outbursts of vortex energy, than due to any automatic Doctor/TARDIS symbiosis or something. Similarly, I didn't get the impression that the "crashing" at the end of "The Time of the Doctor" was anything more than a result of nobody's hand being on the throttle.
Also, strictly speaking, it hasn't been
every regeneration. We didn't see what happened right after the War Doctor's regeneration into Nine. And the Tenth Doctor's abortive regeneration (which does count, as we now know) happened while the TARDIS was stationary. Really, he'd have a lot fewer problems if he just wouldn't regenerate and drive. There oughta be a law about that.
We've known since the beginning that the TARDIS is meant to help regeneration in some way; the first Doctor went back (the long way, it turns out), and as recently as the metacrisis regeneration did Jack order everyone to drag the just-shot tenth Doctor into the TARDIS in anticipation of regeneration (or POSSIBLY just to find cover for the event, YMMV).
That's true, up to a point. The Second Doctor said "it's part of the TARDIS"; I think the original idea was that it was a technologically imposed change, since they hadn't quite settled on the Doctor being an alien rather than a far-future human (Hartnell's Doctor did call himself human on occasion). The second regeneration was induced by the Time Lords, and the third was catalyzed by a Time Lord. The fourth was the first one to happen far from the TARDIS, and it was the most unstable, needing the Zero Room within the TARDIS to stabilize. And pretty much all the rest were in the TARDIS except 8/War, which was under the influence of the Gallifreyan-descended Sisterhood of Karn. I don't think they planned it that way, but it sort of seems to work that way.
In any case, I'm wondering if the Doctor and the machine have become so intertwined over the centuries that a) the TARDIS has lost its ability to help with a regeneration without some consequence, or b) the TARDIS also regenerates to a certain extent whenever her pilot does, possibly behind the circuit panels and not so overtly.
I think that's a pretty huge reach. The only times we've ever seen a TARDIS change coincide with a regeneration were in "The Eleventh Hour" and now in the upcoming season, and both of those can be adequately explained by the damage caused by the Doctor's energy outburst. So there's no need for any speculation beyond that, by Occam's Razor.
My feeling is also that the TARDIS hit her ejection seat button and was trying to save the Doctor from something completely unrelated to her regeneration.
Again, I'm going with Occam's Razor -- the huge explosion that engulfed the entire TARDIS interior a moment after the Doctor was ejected is all the explanation we need. Huge explosions pretty much outweigh any other considerations.
We'll see (not) soon enough, but my feel from watching it has always been that the new team has something specific in mind and is setting the Doctor and ship to be separated for a plot-related reason. The cloister bell was going off BEFORE the regeneration while the Doctor was busy giving his final testament (sic), which has not happened in any other regeneration (though it has happened AFTER every modern one), which might suggest that something else was in the works besides anticipating what the Doctor was about to do.
I don't think there's anything to that beyond a tendency of modern producers to overuse the heck out of the Cloister Bell.
Besides, my impression was that Moffat's work on the episode ended with the regeneration and Chibnall wrote and produced the final scene. So there wouldn't be any connection between Moffat's and Chibnall's uses of the Cloister Bell. Although Rachel Talalay directed both parts, so its use could've been her idea.