Two esteemed sources which say that the universe is both finite in size as well as life.
Finite in size is completely distinct from finite in time. All evidence at the moment points to the Universe being infinite in the future time direction. Your source talking about a cold icy death is talking about an infinite future, not a finite one.
And it has not been settled once and for all, as we do not know the future dynamics of the dark energy field. But it certainly appears very likely that the Universe will continue to expand at an ever increasing rate, and once all the black holes have evaporated and all of the protons decayed (if they do decay), the Universe will indeed be a very cold dark empty place - and will be evermore-so throughout its infinite future.
Regarding the "finite in spatial dimensions" question, we have no evidence that that suggests the Universe is compactifed (as suggested in your references) - I know, I conducted research on looking for this evidence twenty years ago, and nearly everything discovered since then has discounted this option (up to observable scales obviously) - the Universe could still be closed (making it appear like the T>0 section of de-Sitter space), and would then be finite spatially but infinite in the future time direction.
Mathematically nothing finite can exist infinitely apart from an infinite source.
This is not mathematics, it is simply ill-defined. What is an "infinite source"? If the Universe is open and un-compactified, it is infinite and contains an infinite number of anything you care to imagine. This requires no more (and no less) "source" than a finite universe.
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