By making potions.
There are only 81 levels in Skyrim.
No, after everything is level 100 you will be level 81 and you will not be able to level up anymore
There is no first level of Skyrim to be defeated.
Look in kitchens and alchemy shops everywhere.
There is a hard cap of Level 81. This is determined by people using a cheat to put all their skills to 100 and seeing what level they become. When all skills are 100, there's no way to level further, so 81 is the cap.
The max alchemy level is 35 which heals 625 HP.
It is actually really easy. All you have to do is get a lot of leather strips and iron bars. Then keep on making iron daggers till you are level 100. It shouldn't take long.
No, Level 81 is the highest that player characters can ascend to.
You can reach level 81 in Skyrim I think but that will take an awful long time, and to be honest, who would be bothered.
You don't "make" the alchemy labs, in the normal sence. You can either use the ones in the alchemy shops (where you buy supplies in almost any town) or it's something you can buy as an addition for any house - other than Whiterun - from the same person you bought the house from.
Health and Mana potions will heal 100 HP and MP (respectively), though it is possible to increase this value with Alchemy training. The Alchemy profession is taught by Reens in her aforementioned shop. In order to increase your alchemy level, you must gather reagants, bring them back to Reens, and complete a "transmutation circle." Though it may take you a while, the process is not as difficult as it sounds. A more in-depth process can be found in the Reens' Shop pedia entry. For each alchemy level you gain, your health potions will heal 15 more HP; you can increase the potency of your health potions up to 600%. A guide about alchemy can be found here: How To Become A Successful Alchemist
The only place you can train your alchemy level in Maple story is by going to Ardentmill which is the crafting town. You have to make potions to increase your level.