Examples of High Magic Settings

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Isn't Gandalf like the weakest member of his race? I'm sure the strong Maiar can do those things (except open a gate to another plane because there are no planes in the LOTR universe).

Gandalf isn't the weakest, he was the second-highest ranked of the five wizards. The Istari were banned from using their full power to confront Sauron directly, they were intended to help and encourage the people of middle-earth to fight him instead, and also they were limited by being bound to a mortal body. The only time Gandalf ever used his full power as far as I know was in the fight with the Balrog.

But yes, while many of the spells in D&D are never mentioned by Tolkien, the Ainur did have vast powers in their original forms, so much so that the Valar swore never to use their full strength again after they nearly destroyed the world by doing so.

Well, when I say high magic setting I was thinking about a setting where even the lower class common folk have access to magic that provides them with a level of comfort similar to the modern times, and a setting where soldiers use spells instead of bows, every chef knows prestidigitation, things like that.

The closest to this that I know is the Netherese empire in Forgotten Realms, the mythallars allowed them to make magic items without needing to expend XP (although they only worked within one mile of the mythallars) which meant that they became much more common than elsewhere.

To a lesser extent, in the later FR setting Halruaa, Nimbral, Evereska and Evermeet had a substantial portion of the population who had some rudimentary casting ability (as described in the Magical Training feat from PGtF). In Shining South it says that one-third of the population of Halruaa have some sort of magical ability, mostly Wizard spells.

Edit: oh, and in the Discworld books low-level magic items become very common, being used to make things like cameras and personal organisers. Also golems provide continuous power for certain types of machinery.