People equally stupid to this thread's author could argue that Mihawk is a small frog who went after the Government's protection and lived in Paradise, only takes action against the likes of Don Krieg, ask for postponement versus a mere commander, helps Crocodile in order to stay away from trouble and is surpassed in willpower by Buggy while expressing his concerns about getting involved with a conflict between the emperors.
But again, arguing that would be stupid because it's more about Mihawk's character than his strength. That said, regarding your "points":
- Never left his mother's territory
- Acts tough infront of his weak brothers
- He's afraid of his mother
- Got his bounty because of his mother's reputation ( nothing indicates that he left WCI )
- Think he is invincible because he never challenged someone strong
-He did, he has been travelling the world since the days Big Mom was part of Rocks. Also Big Mom comments on how long it's been since the three Sweet Commanders were togheter at the same place, implying they do travel (just like every other emperor crew).
-And? He acts tough in front of his siblings because that's a way to protect them. He's still the stronger of them all, anyways.
-Yeah, of course he's afraid of his mother, why wouldn't he and why would this be related to his strength? Linlin isn't just his captain: as you said, she's his abussive, chaotic, unstable mother. Their relationship is obviously different than any other and Linlin's issues don't simply affect him but his siblings, which is what he fears the most.
-He obviously got his boundy for something more than just his mother's reputation considering the fact his twins are only 300 million.
-He doesn't believe he's invincible, his siblings do; it's a facade to protect them but the real Katakuri is the merienda one. Doesn't stop him from having a power only the mightiest have according to Rayleigh, an ability rarer than advanced conqueror's (confirmed advanced conqueror's users: 9; confirmed future sight users: 4), and awakening on top of it (12 confirmed awakened users, but 5 of them lost control of themselves so....).
A pointless thread, in conclusion.