For the first example it nearly always must be a noun in that position. The article "the" is followed by a noun phrase. If the noun phrase consists of just one word, that word must be a noun. It might be possible to create some artificial sentences with an adverbial "the":
The more carrots he eats, the better can he see!
Now, in this sentence "the" isn't an article, it's an adverb so this doesn't technically fulfil your requirement, but it might be close enough.
In the second case, it is quite easy to have,
Can the cats play tennis?
Can most cats play tennis?
The word "the" or "most" are determiners and not verbs.
You might be able to achieve this with a rhetorical inversion:
In the hole can the cats hide.
This is an example of a rhetorical technique called Anastrophe, changing the normal order can be done for emphasis. This is not a good example of anastrophe, I've adapted it from a well-known example "In a hole in the ground there lived a hobbit" (Tolkein)