My impression from the Pali suttas is that ...
The Buddha did escape samsara.
That escape (or liberation), nibbana, is understood as permanent -- perhaps contrasted with a Mahayana doctrine, which views the nibbana of arahants as temporary (?) -- and contrasted with earlier stages of enlightenment, which include some glimpse (but only glimpses) of nibbana.
I'm not sure about the dtstinction between "permanent" and "timeless". Dhamma for example is described as akalika (without time). I don't know what Pali words are used to describe nibbana as permanent ... maybe I've misremembered or misunderstood, and instead of "permanent" it's more a word like "total" or "complete" or "final".
I think it's permanent is that the underlying roots have been severed: like if you cut the top of a tree that might grow again, but if you cut its roots then that doesn't happen; or if you smother a fire it continues to smoulder, but if you remove the fuel it won't.
I think that an aspect of the Buddha's nibbana was the ability to easily attain jhana states, e.g. to escape physical pain --- and, to escape mental pain, non-attachment (to avoid any sense of loss or yearning) as a result of right view (seeing everything world as impermanent and not as an object of desire); plus entirely ethical intentions and behaviour (therefore no sense of remorse).
Death is called Parinirvana a.k.a. cessation of the aggregates -- which is also I think "nibbana without remainder", in that after enlightenment but while the person is still alive, remaining karma is coming to fruition though no new karma is created.
No "regret", I'd presume -- because doesn't the word "regret" imply attachment or remorse? ... unless perhaps you might infer or project regret from the fact that he was still trying to help his followers gain enlightenment themselves: that might be better described as "compassion" though, or something like that.
As for whether the Tathagata exists after death, that's one of the canonically "unanswered questions".
I guess perhaps you'd say yes, if you equated the Tathagata with the Dhamma -- the Maha-parinibbana Sutta says,
Now the Blessed One spoke to the Venerable Ananda, saying: "It may be, Ananda, that to some among you the thought will come: 'Ended is the word of the Master; we have a Master no longer.' But it should not, Ananda, be so considered. For that which I have proclaimed and made known as the Dhamma and the Discipline, that shall be your Master when I am gone.
... is that like what the Mahayana means when it talks about the Dharmakāya?
There's a description of the Buddha's "death" in the Pali Maha-parinibbana Sutta -- scroll towards the end: the beginning of "Part Six: The Passing Away".
The Pali suttas generally only talk about one Buddha, not all "Buddhas" in general.