Hello brother, "haut" is masculin, we say "le haut", just like "le bas, le côté, le milieu...". You can say "je monte en haut, je vois le haut de la tour".
"L'haut" makes no sense, it would sound like "l'eau" and nobody would understand. And "la haut" doesn't exist, but maybe you meant "là-haut" which is more like "up there", up in the sky. example : je vais monter là-haut.
some words work like "homme", like "l'hérault", "l'honnêteté"...other ones work like "haut" : le héros (notice that it sounds exactly like hérault but works differently), la haie, la hyène etc.
regarding capitalisation of names, Homme means human, like "les Hommes ont créé Dieu", and "homme" with little h refers to men, boys. So for example, "human rights" would be translated by "les droits de l'Homme".