Is it tennessINE because it has 117 electrons or because it behaves like a halogen? ... Or will all current and future elements that obeys certain numerical rules be called halogens no matter how they behave chemically?
Yup, that last bit is the most correct. The math and physics behind how electrons form orbitals meant that scientists already knew that Element-117 was going to be one-electron-shy of filling that outermost shell of valence electrons (and they knew it for decades before they were actually able to synthesize it in a lab).
[Tennessine has a -INE name like all the other halogens. Did they test whether it actually acts like 'a halogen'? Or is it simply assumed to behave like the rest? Or does it get to be an honorary halogen just because its electron configuration shoehorns it into the same column as others?]
The half-life for the Tennesine-293 isotope are only about 25ms (and 51ms for Ts-294). That is only really enough time to see how the atom continues traveling through the particle accelerator and tracking the pieces after it decays (to make sure all the masses add up correctly). It is not enough time to do the kind of "real chemistry" of testing whether Tennessine "acts like a proper halogen".
So, is it assumed/calculated to behave like the other halogens? Well that depends what you mean exactly. Fluorine, Chlorine, Bromine and Iodine are all "Non-Metals" whereas Astatine might be a "Metaloid". Tennessine is likely to be even more-so metallic than Astatine - being either a definite "Metaloid" or maybe even a full-on "Metal". One could possibly argue that being a metal disqualifies it from being a 'proper' halogen (while still being Group-17)... but Astatine has been skating by on its shaky credentials since the 1940s so it's probably fine to let them both keep the -INE names. Also, it's not like folks are all that strict about differentiating 'proper' Chalcogens or 'proper' Pnictogens, so why start now with 'proper' Halogens? Sure, it is a bit of a subjective judgement call, but most of the chemistry world is on board with it.
... I questioned how elements discovered in the future might or might not be classified as halogens, especially if an "island of stability" is ever found and even more elements are discovered.
This might be trickier to answer. Part of the reason why scientists were so easily able to just fit Tennessine into the modern table was because Period-7 follows the same pattern of orbitals as Period-6. Period-6 elements mostly have stable isotopes so this essentially gave a mostly-complete template for what the Period-7 row should also look like. (And it worked like a charm as scientists filled out the gaps in the Actinide series and the gaps in the heaviest transitions metals from like 1930 to now.)
We have zero examples of elements in Period-8, so we can only hypothesize that there is a g-block of orbitals that gets added on. Assuming that is all correct, the expected number of s/p/d/f/g electrons needed to fill a Period-8 shell means that the next 'halogen' element would need to have an atomic number of about 167-171... which is a long way off if only fleeting unstable isotopes are left to discover.
If there is an "Island of Stability" then we might luck out and be able to learn more about the electron configuration of Period-8 enough to either keep the g-block idea or scrap it for something better based on that new evidence... but either way you look at it, another new halogen seems unlikely anytime soon.