Strings of sound can be given an interpretation in at least two ways. The usual method, whereby we typically define "possible interpretation", is "strictly using the rules of the grammar in this language". Another method – the one typically used by the overwhelming majority of humans – is "using any and all evidence and methods available". The latter method is extremely useful when called on by individuals attempting to communicate, who do not have a common language: it allows one to be understood, even when you massively violate the rules of grammar.

There are many pre-compiled strings that are used in language differently from what you would expect given literal semantics, for example "pot calling the kettle black" is literally an absurdity. Moreover, the actual interpretation of the utterance cannot be related by rule to word meaning and principles of compositional semantics.

The answer to your question depends on having an explicit theory of lexical meaning, syntax, and semantic composition. It presupposes that syntactic distribution is blind to semantic properties – a possible but not logically-necessary position. All attempts that I am aware of to compose the meaning of strings from the meaning of the components (including abstract "thematic role" marking) are very general – they say that you can can combine "P", "x" and "y" to derive the proposition "P(x,y)" as well as "P(y,x)". In such compositional theories, all well-formed sentences have a semantic interpretation. But: not all such propositions describe actually-possible states of affairs. Avicenna's defense of the law of non-contradiction ("Those who deny the first principle should be flogged or burned until they admit that it is not the same thing to be burned and not burned, or whipped and not whipped") reminds us that we can conceptualize things that do not exist, and that contradict the nature of the universe.

In other words, it depends on what you mean by "mean". If you think meaning is about actual states of affairs in the real world, then it's easy to construct grammatically well-formed strings that have no meaning because they don't describe facts. I just think that is the wrong theory of meaning.

Yes, there are indeed different kind of semantically meaningless or anomalous sentences, and those different types can be distinguished in psycholinguistical experiments, e.g., by using the EEG signals N400 and P600.

I think the common term would be non-idiomatic, idiomatic here not referring to idioms like "kick the bucket", but to the natural ways a language is spoken.

Specifically answering your request for a lexicon of such words, this webpage includes a link to download multiple lists of function words.

"Although" appears in its list of English conjunctions. The class of function words are defined by this source as including Auxiliary Verbs, Conjunctions, Determiners, Prepositions, Pronouns, and Quantifiers. The OP may choose to disregard some of these sub-categories as not applicable to your purpose.

The panda eats, shoots and leaves.

The syntax is correct: it relates an observation of a panda eating before shooting and leaving. However, the misplacement of the comma makes the sentence semantically incorrect, as the intention of the sentence should be that pandas eat shoots and leaves, not that this panda was shooting. (No offense to the Kung-Fu Panda, who may actually shoot.)

Assuming that, 100% of the time, there will be a sentence that is semantically correct, and another that isn't, you can just split the type0 and type1 sentences into 2 different examples and classify them individually, e.g.:

id,type0,type1
0,He married to a dinosaur.,He married to a women.
1,She drinks a beer.,She drinks a banana.
2,He lifted a 500 tons.,He lifted a 50kg.

Becomes:

id,sentence
0,He married to a dinosaur
1,He married to a women.
2,She drinks a beer.
3,She drinks a banana.
4,He lifted a 500 tons.
5,He lifted a 50kg.

However, this won't work if your data contains records where a sentence is slightly less stupid than the other, i.e. there's the actual need to compare both sentences.