...No. Ichigo was faster than Byakuya in their fight, but Tsuki wasn't. Tsuki knew where the attack would land/hit before Byakuya even thought about it, that's why Tsuki could avoid all of them, beside the one he never have seen/experienced.
After cutting his sword (and creating a bond ;P), Tsuki described their relationship as training partners, although you don't see that much of a upper hand from a training partner, rather it looked like and old master (Tsuki) vs his pupil (Byakuya), so even though Kuchiki's stats were higher than Tsuki's, his experience allowed him to have the upper hand through out most of the fight.
It indeed looks like Riruka entered Rukia, it's possible that the new improvement allows her to enter "cute" things.
I reread the part were Byakuya used the thrusting palm strike to finish Tsuki and I think I know what he did with his bankai petals.
You need to first think what his bankai is, and that is many swords in the form of petals that fly at high speed. By catching a few of them, they still can move, but if something is holding them then they can pull it with them, on the other hand when something is blocking them then they push it or pierce it.
What Byakuya in fact did, was that he caught some of his blades and made them move in Tsukishima's direction while he was holding them, so then they pulled him, allowing him to perform an instant dash jump (perhaps he added a shunpo to it), but that's not all, normally the speed is high but the mass of a few petals isn't high, but added with Byakuya's weight, it got enough power to penetrate Tsuki with his bare hand.
Theoretically, using his petals, Byakuya could fly like a Superman

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