This is actually quite a good question and I hope rioters can fill us in on more details.
He probably takes his gauntlet off in Demacia so they don't suspect him. He obviously went there to find his parents/make them find him and maybe he heard about the world runes and wants to find them?
The national hero part doesn't make sense currently, there's a chance it's all in his head or it's a lie he made people believe. He probably just wore fancy clothes and they were like, ok makes sense. There's also a chance he is a hero by defeating noxians in some remote demacian village but he doesn't look like a warrior(which would make them suspicious of him beig a mage) so it would be interesting to know what he did to become a hero.
As for Sylas, we know he can see magic so he can see right through jarro lightfeather and obviosuly sees his magic. As a fellow mage he probably would want Ezreal to join his cause unless he disliked him because Ezreal helped the mageseekers or something (or some lux love triangle).