Even in game mode, I can't hit Tidus's overdrive. I think this will make lightening dodging impossible. Will my input lag cause me to have problems with anything else in the game?

The chocobo racing if you're after Tidus's ultimate weapon probably, and the butterfly stuff for Kimahri's. You'll probably find the temple with the moving platforms a pain too. Other than that, twitch could probably play it.

Maybe your monitor can't lower enough delay even in game mode.

if your using an LG tv game mode is terrible - but renaming the HDMI input to "pc" eliminates the issue

also, pc mode on heaps of brands has less lag than gaming mode

There is a spot in the thunder plains where you can repeatedly dodge lightning and it will always come at a certain time. once you get the hang of when it comes and when you have to press the button, do it over and over and your brain will get into a pattern. put on some music and it'll be ok.

Then again, I played on the Vita so I'm not sure if the same issues is there since the screen is built in.

My TV is a westinghouse, and it has a really small menu. I'm not even sure if game mode does anything besides adjust stuff like brightness, contrast, color, and hue.