Implicit "that/which is/are" in nonrestrictive relative clause

Today I stood in the train, [which is] still better than walking. [had to stand, because there were no free seats]

May be a realization of your first proposed form. However it is grammatically incorrect to begin with unless the comparison is focused on the closest possible reference, the train, which would be a bit sketchy. It can't be comparing the verb, which would require an adverbial comparative instead. It can't be comparing the subject or the whole clause, although that's probably how most will parse it anyway, because the distance between referent and referee is large, but not too large. We'd have many options to change the sentence, but omitting "which is" does not seem to be one, precisely because better would appear to attach to train and so you'd say stood worse walking, which does seem a bit of a peculiar notion. ... Oh, sorry, I meant "... and so "stood worse walking" would be the confered statement, which ...".

Today we have soup [which is] spicier than yesterday's

We generally eat our soup spicy. We eat soup spicy. On mondays we eat soup less picy than other days, because we try other flavours, too.

This is totally OK, but works best because eating spicy is a collocation. If you said we eat soup, which is spicy it does not yet exclude bland soup, but we eat soup spicy does, even if only implied, because the word order allows combining more adverbials we eat soup spicy out of principle.

That said, compare that to

I have a dog big as a horse

* I have a big as a horse dog?

* I have a dog. This situation is as big as a horse?

I have a dog, which is as big as a horse.

Well sure it's the dog that is referenced. That is eminently clear. There is no need to use a specific pronoun or other determiner. And while "I have a dog big" would sound peculiar, it's quite normal with specific meanings of have,"I'll have my steak rare" (never which will be, though I'll have me a rare steak or I have the rare steak exist).

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Your third form I do not understand. Mary has a dog bigger than she/herself shows that the syntax is confusing across the board, but people bicker over Mary and he/him, too, all the time.

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The later two forms work because participles are already adjective like, and "Mary has a dog [that is] biting the mailman at every chance" is as acceptable as "The dog biting the mailman belongs to Mary". Which is a short way to a point free normal form Dog bites, belongs, barks, .... Inserting which before every new clause would not help much. which is actually a question word--that is not hard to tell from the onset wh. You are rarely implicitly asking "I had to stand on the train. Well, what else would have been better than walking?*".