Does the story get better? Yes. Pretty much everyone agrees that the story gets better later on. When it gets better, and how much better is open to debate(im of the opinion it was mostly "meh" until patch 2.something but has since picked way up).

Does the combat get better? A level 1 lancer has the same GCD as a level 70 dragoon if theyre wearing the same gear. The level 70 version has 2 combos to weave around something like a dozen oGCD skills and various buffs and resources to manage(the level 1 version has true thrust and autoattacks). The level 1 version is fighting enemies with only basic attacks and maybe 1 telegraphed aoe, the level 70 version is fighting some enemies with fairly intense and extensive mechanics(along with their henchmen with mostly basic attacks and maybe 1 or 2 telegraphed bigger moves). So while it definitely gets more involved, better is more subjective.

Yes, the story will continue to force you to do dungeons and/or trials every few levels all the way up. DPS ques can be rediculously long. This can be a hassle. There are work-arounds, but some instances are unavoidable.

And quests. Im not sure what exactly you were expecting. Every MMO ive played had a ton of fairly simple fetch and kill quests. Some of them also had a few more ambitious quests mixed in, but they were always the exception and not the rule. Same is true here.