I have heard that many things (quests/items/etc.) in Skyrim will not available to low level player.

Some items even have different stats/ attributes, depending on player's level when they first acquire those items.
NPCs/Enemies/Followers' stats will be set to player's level when player first meet them, and will never improved.

These make me think I might screwed up my gameplay by exploring too many places, doing to many quests, early in the game without max leveling my character.

How much contents will I miss if, let's say, I explored most parts (beside dungeons) in Skyrim before level 10?

And what is the lowest level I have to reach if I want to have access to all items and quests?

I'm not aware of any quests or items that aren't available at low levels, some items will be not as impressive as they could be but they'll still exist.

This
Even then monsters and items stop levelling at 50th level and max level is 81st so you defnitely don't need to max out your character to experience everything

Daedric quests have a level requirement.

I doubt you'll "miss" any content. It's going to be very difficult to play through the game without hitting at least level 20 (the highest requirement for all but one Daedric quest... one has 30), especially if you were doing lots of exploring like you said.

My first character I'm trying to get all the way up to the max level, and, honestly, it's not as much fun as playing a character that naturally "caps" due to your playing style. My thief is a sneaky bastard who is great with a bow... but now I'm to the point where all that's left to level up are 1H, 2H, Heavy Armor, and block... so after playing 50+ levels one way, I am running around like a barbarian. And this is after playing levels ~55 - ~70 like a mage to get all the magic skills to 100. It sort of breaks the immersion.

I'm doing it with my first character to make my first playthrough last as long as possible, in hopes that a few more patches will come out before I start another character, so that I will experience fewer bugs. I plan on playing my next character more "pure," meaning I'll pick one or two skill trees and max them out, and just let the max level fall where it may.