To answer some of these questions.
It’s pretty clear that Godzilla won the fight. Even though he didn’t kill Kong (although Kong did come close to dying), Godzilla beat Kong down and asserted his dominance over him (which is what mostly happens with animals in nature). At the end of the movie, Kong drops his axe when confronted with Godzilla, showing that he submits and doesn’t want to fight again (during the confrontation, Godzilla also motions towards the axe, as a way of telling Kong to “drop it”, which indicates that Godzilla has no desire to continue the fight either).
It makes sense to focus on Kong as the main character as he is more human-like, which inherently makes him more relatable and gives him more personality than Godzilla, especially given Toho’s recent trend of wanting to portray Godzilla with little to no personality/motivation (apparently Legendary really had to push for Godzilla's expressive scenes in GvK). Kong is also much more sympathetic as he's portrayed as the underdog in the fight and continuously goes through hell during the events of the movie, when all he really wants is to have a peaceful life and a new home. The film also emphasizes that because he’s presumably the last of his kind, he’s extremely lonely and wants to be part of a family (which is why he's so caring towards Jia and why he immediately enters the Hollow Earth at the possibility of finding others of his kind there). Because of this, it’s easier to root for Kong in this story, rather than Godzilla. While the film focuses more on Kong as the main character, it doesn’t diminish Godzilla’s role in the film, he still plays a huge part in the story.
The axe was made by the ancient Kongs that lived in the Hollow Earth, using the skeletal remains of other Titans (specifically the Gojira species). It may initially seem odd that a species of giant apes would create axes, however real life species of great apes have been observed using tools, with accounts of chimpanzees and bonobos crafting makeshift spears and using very primitive stone tools and sometimes bones (similar to early human ancestors). So it's not too much of a stretch that a more advanced fictional species of giant ape would create complex tools to help in their daily lives. Plus building weapons would help to give the Kongs an advantage over other Titans, since they lack energy attacks (like Godzilla’s atomic breath) or other supernatural abilities.
As far as Mechagodzilla's plotline goes, there's no indication that it's focused on Kong aside from APEX wanting to use him to guide them to the Hollow Earth energy source. Most of the focus is directed towards Godzilla, with APEX specifically modeling their creation after him so that they can rival his power. APEX feels threatened by the Titans’ existence and they desire to be stronger than them so that humanity can remain the dominant life forms on the planet (which is why they're called "APEX" in the first place), and given that Godzilla is the "Alpha Titan'' and therefore, the dominant life form on Earth, they want to overthrow him in order to claim that title. When Mechagodzilla is later possessed by the consciousness of King Ghidorah, he immediately goes for Godzilla, coming close to killing him until Kong intervenes. Even during the main fight, Mechagodzilla is more focused on Godzilla, aiming most of his attacks at him, while Kong is viewed more as an annoyance that Mechagodzilla initially tries to shake off. It isn’t until Kong starts doing a lot of damage that Mechagodzilla finally turns his attention towards him and actually tries to kill him.
Lastly, Godzilla being the main focus of a film doesn’t necessarily mean that he’ll have more screen time. Both Godzilla (1954) and Shin Godzilla have him as the sole focus, yet he still has minimal screen time in those films, with a lot of moments where he is either absent or inactive, and the Polygon anime trilogy barely used him despite the fact that he was supposedly the main focus. Godzilla’s screen time in the next Toho film will mostly depend on how they want to use him, so it's possible that he might have more screen time, but it's also possible that he won't (and given how Toho has been portraying him in their recent projects, it's very possible that it might be the latter).
"Neither beast nor man. Something monstrous."
