I find it strange that you are labelling ten/tie/.. as with gender. I don't know Slovak, but I'd expect it is the same as in Polish that the gender is on the subject. E.g., stôl is stół in Polish, and "męskorzeczowy", so masculine. "Ten stół" or "te stóły" -- te or ten is neither feminine nor masculine.
These demonstrative particles themselves don't have gender since they are not nouns, but they have a gender-based variety, and must pair correctly with the nouns by gender.
It's similar to la and le in French. You cannot say "Vive le France"; it has t obe "la France".
They are used as helpers in communicating the gender of a noun. If we say "ten stôl", it reaffirms that the noun is masculine. "tá stôl" is ungrammatical.
Other words are like this. E.g. interrogative wh- words: "ktorý muž?" (which man?) "ktorá žena?" (which woman?)
