How can I brew Potions of level 2 and the Potion of Invisibility? Are some of their effects available with the beacon or there really are potions of lvl 2?
There are two kinds of level II potions. Adding glowstone to a potion will give you a stronger version of that potion while adding redstone will give you a potion with the same properties but with an extended duration. Some potions like the night-vision potion and fire resistance potion get no benefit from glowstone.
As for the potion of invisibility, you can brew that potion by taking a potion of night-vision and adding a fermented spiders eye. Generally adding fermented eye to any potion gives its opposite, such as potion of swiftness to potion of slowness or potion of instant health to potion of harming.
Beacons can also provide a large number of buffs that can also be given by potions. The buffs shared between the two are strength, speed, jump boost, and regeneration. The only buffs exclusive to the beacon are haste which increases mining speed and resistance which reduces damage. The beacon require a lot of spare materials, has a limited range, and can only have one of the buffs at level II or one level I buff and regeneration.