What would it mean for a color to smell?
There may well be "smelly colours" (or at least colourful smells), it is called "synaesthesia". Smells and colours are not real, they are our perceptions of aspects of reality. There is nothing to stop someone experiencing a colour as having a smell, and a quick web search suggests there have been recorded cases of smells being perceived as having a colour.
The reason "saying that a color is smelly doesn’t seem to be as contradictory in a way that a square triangle does" is because it actually happens, whereas for the definition of a triangle precludes it being square (at least for Euclidean geometry - who knows what mathematicians get up to!). Unlike smelly colours, it is ruled out a-priori by definition.
Or that “we can’t know if a smelly color doesn’t exist. It is
certainly possible since it can’t be disproven”. You’d be laughed off
the face of the planet if you did.
Those laughing would feel pretty silly when they found out about synaesthesia! ;o)
If you had asked most scientists 400 years ago whether it was possible to have a transparent vertebrate, it is likely that they had never seen any evidence of such a thing and they wouldn't have seen evidence that it was possible. Of course they couldn't have been disproven it either. Yet such creatures exist.
For starters, one can’t imagine a being that is immaterial. Can you?
Well, actually yes, and I am far from being the only one, as you self-answer:
And yet, there have been thousands of books, millions of pages, and
full encyclopedias written about this seemingly nonsensical concept
To use one of your other examples:
We don’t remain agnostic to the existence of a crying leaf or a living stone.
I can imagine silicon based life, and an immobile silicon based life form (the equivalent of a lithops) would be a "living stone".
So what does all this mean? Carl Sagan writes in one of his books (but similar statements had been made previously)
Keeping an open mind is a virtue—but, as the space
engineer James Oberg once said, not so open that your
brains fall out.
I fully agree with this as I am skeptical by nature and ruling anything out is not very skeptical.
Relying on falsifiability for all questions seems like naive scientism - the belief that science is the only or best route to knowledge on all topics. However, while Popper argued that we can't prove anything, only disprove, the Quine-Duhem thesis shows that we can't unequivocally disprove anything by experiment/observation either.
So this seems to support Sagan's maxim - we should keep our mind open enough that we consider arguments and evidence, while at the same time remaining skeptical.
The real problem is that things are either possible or they are impossible, the difficulty is that we don't know which. So if we can't show something is impossible there is an epistemic possibility that it is possible in reality. Arguments often confuse epistemic possibility with possible in reality. They are not the same thing at all.
As an example, "faster than light travel is possible", is not a falsifiable statement. We can argue based on theory that it is impossible, but we can't demonstrate it. Most people would accept that it is false, but it would be lacking in proper scientific skepticism not to consider there is a possibility that it is true. Einstein may have been wrong (or at least incomplete) just as Newton was before him.
Similarly, "there exist anti-matter galaxies outside the visible universe" isn't falsifiable either, and we have no evidence (except from theory) to support it. Should we refuse to entertain the possibility simply because it is unfalsifiable?
Just one last example. The existence of Black Holes was first proposed by John Michell in 1783. He suggested there may be "dark stars" in the night sky that we can't see. That is not a falsifiable hypothesis, while it would be possible to detect some "dark stars" if they had a visible pair as part of a double star system, however that doesn't rule out the possibility of unpaired dark stars in intergalactic space. Does that mean we shouldn't accept the possibility that the hypothesis was true (because it was, but not quite in the way Michell thought)?
BTW, as you may have worked out, while I appreciate falsificationism, I am not a hard-line Popperian.