Have Buddhas escaped samsara?

This is what Archarya Nagarjuna has to say on this topic:

25.17 Having passed into nirvana, the Transcendent Lord
Is neither perceived to be existent
Nor perceived to be nonexistent.
He is neither perceived to be both nor to be neither.

25.18 So, when the Transcendent Lord was alive, he
Was neither perceived to be existent
Nor perceived to be nonexistent.
He was neither perceived to be both nor to be neither.

25.19 Cyclic existence is not the slightest bit
Different from nirvana.
Nirvana is not the slightest bit
Different from cyclic existence.

25.20 Whatever is the limit of nirvana,
That is the limit of cyclic existence.
There is not even the slightest difference between them,
Or even the subtlest thing.

25.21 Views regarding his status after his passing; extremes, etc.,
And views regarding the permanent, etc.,
Are grounded upon nirvana, the final limit,
And the prior limit.

25.22 Since all existents are empty,
What is finite or infinite?
What is finite and infinite?
What is neither finite nor infinite?

25.23 What is identical and what is different?
What is permanent and what is impermanent?
What is both permanent and impermanent?
What is neither?

25.24 The pacification of all objectification
And the pacification of all fabrication is peace.
No Dharma was taught by the Buddha
At any time, in any place, to any person.

Warning:

If you take anything said above as indicating that nothing exists or that this line of thought refutes the Four Noble Truths, Karma, Rebirth, the Dharma, the Buddha, the Path and the Fruits, etc., ... nothing could be further than the truth! Consider this warning a handrail to lean on if you take the above to indicate nihilism:

By a misperception of emptiness
A person of little intelligence is destroyed:
Like a snake incorrectly seized,
Or like a spell incorrectly cast.

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.

According to the Mahaparinibbana Sutta:

And soon after the Blessed One had eaten the meal provided by Cunda the metalworker, a dire sickness fell upon him, even dysentery, and he suffered sharp and deadly pains. But the Blessed One endured them mindfully, clearly comprehending and unperturbed.

After Nibbana, the Buddha could indeed experience the sensation of pain (as he did in the case of dysentery), but he nevertheless does not suffer from it. Hence, suffering had indeed ended for the Buddha.

According to Itivuttaka 44:

This was said by the Blessed One, said by the Arahant, so I have heard: "Monks, there are these two forms of the Unbinding property. Which two? The Unbinding property with fuel remaining, & the Unbinding property with no fuel remaining. Footnote And what is the Unbinding property with fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. His five sense faculties still remain and, owing to their being intact, he is cognizant of the agreeable & the disagreeable, and is sensitive to pleasure & pain. His ending of passion, aversion, & delusion is termed the Unbinding property with fuel remaining.

And what is the Unbinding property with no fuel remaining? There is the case where a monk is an arahant whose fermentations have ended, who has reached fulfillment, finished the task, laid down the burden, attained the true goal, ended the fetter of becoming, and is released through right gnosis. For him, all that is sensed, being unrelished, will grow cold right here. This is termed the Unbinding property with no fuel remaining."

Commentary (Thanissaro):
With fuel remaining (sa-upadisesa) and with no fuel remaining (anupadisesa): The analogy here is to a fire. In the first case, the flames are out, but the embers are still glowing. In the second, the fire is so thoroughly out that the embers have grown cold. The "fuel" here is the five aggregates (see the Glossary). While the arahant is still alive, he/she still experiences the five aggregates, but they do not burn with the fires of passion, aversion, or delusion. When the arahant passes away, there is no longer any experience of aggregates here or anywhere else.

After attaining Nibbana, but being still alive, the Arahant or Buddha still could experience the sensations of pain because the five aggregates are still operating. But he will not suffer from it, or have passion, aversion and delusion. Meanwhile, after death, the five aggregates no longer operate anywhere.

In any case, suffering ends completely after Nibbana. The Buddha does not suffer from disease and death, although he may experience it.

Regret requires aversion and ignorance. It's ignorance that makes one think that there's anything worth in Samsara to miss. Enlightened beings(Arahaths and Buddhas) have uprooted all defilements. So it's technically impossible for them to feel regret or sadness.