It might sound like that because the S of Das goes fluently into the Sch of Schwein (DaSchwein), so there is not really a stop between them. This is what might sound like "der" to non-mothertongue speakers, but it is actually a short "da-".
You can also say it with a stop "Das - Schwein", but nobody does that in everyday speech. Maybe a TV or radio moderator would.
Is “swine” a neutral word to refer to pigs to you? When I hear it, 90% of the time it’s used as a curse word for somebody despicable, and when does refer to actual pigs, it’s usually a) in a fairly archaic context and b) used as a collective plural noun (I don’t remember I ever hearing anybody use this word for a single pig, and it is often even cited as an irregular plural for “sow” (here for example), though etymologically speaking they are separate – albeit related – words). So to me swine feels off here. But if you’re a native speaker and it feels natural to use the word here, go ahead and report it next time it comes up.
If you mean this animal, we usually call that Wildschwein in German. I guess in theory Schwein could be used to refer to them, too (after all a Wildschwein is technically a sort of Schwein). but in practice people rarely do that and when I hear Schwein I think of the domesticated variant (the full name of which is Hausschwein) rather than a wild boar.
I, too, clearly hear "Der ..." The moderator says that "Der" is an impossibility, which is fine if one actually knows this, but I am just learning, and by definition, do not know it. Surely it is not beyond the capability of the technical staff to make the spoken words at least sound unambiguous. Further, from the comments above, this pronunciation has been an issue for years. As a result of this clumsiness, I have not learned that "Schwein" should be "Das", instead I have learned that it might be "Der" or it might be "Das", and I shall dither and swither every time I encounter "Schwein".
Klein can be used for pretty much anything with a physical shape (and even figuratively for many abstract things, e.g. ein kleines Problem “a small problem”). What is different for people is that if you describe a person as klein, it means “short”. But that’s just because we conceptualise people’s height as being groß or klein and English doesn’t.
Don’t think of klein as having the meaning “short”; it doesn’t. It means “small”. Only in the context of describing people’s heights, English conceptualises this as “short”, while German conceptualises it as klein. Same for groß: It only means “big, great”, but when describing people’s height, you have to translate is as “tall” because English uses that adjective in such a context, not “big” or “great”.
No, just for people. Short would mean something different in English for animals as well: Whereas for people it describes height, for animals it would sound very strange but if used at all it would refer to the length from snout to hindquarters (or maybe tail end) – the same as German kurz. To talk about a (legged) animal’s height you would normally go the same way as German goes for both animals and people and say “small”.
You can always keep in mind that the most important meaning of "klein" is "small, little".
For children, for instance, it also often means "very young", just like "a small child" or "a little girl/boy" in English. Of course, that often goes hand in hand with being short in height.
For grown up people, it is short in height. "Der Mann ist klein." = "The man is short. / The man is small."