(Part 2)

To a limited extent it does resemble the language of literary fairy tales.

There is a conscious "storyteller's voice" in there. It reminds me of the style of Nathaniel Hawthorne's "Tanglewood Tales," or John Ruskin's "The King of the Golden River," or Rudyard Kipling's "Just-So Stories."

The characters in the novel, particularly the nobility, the kings, the wizards, and so forth, speak in a higher or more formal register, and--unlike present-day speakers--do so while avoiding Latin-derived words. They also avoid contractions (can't, wouldn't, won't).

Gandalf:

'Yes, you saw him for a moment as he is upon the other side: one of the mighty of the Firstborn. Lie is an Elf-lord of a house of princes. Indeed there is a power in Rivendell to withstand the might of Mordor, for a while: and elsewhere other powers still dwell. There is power, too, of another kind in the Shire. But all such places will soon become islands under siege, if things go on as they are going. The Dark Lord is putting forth all his strength.

'Still,' he said, standing suddenly up and sticking out his chin, while his beard went stiff and straight like bristling wire, 'we must keep up our courage. You will soon be well, if I do not talk you to death. You are in Rivendell, and you need not worry about anything for the present.'

No contractions--he doesn't say "if I don't talk you to death" or "you needn't worry." And I don't think there are any Latin-derived words in there, except "present."

Part 1: My answer is "no--maybe a touch here and there, but no."

Let's ignore the languages Tolkien invented (Elvish, etc.) For the moment, let's ignore the quoted dialog the character speak.

The narrator's voice is not archaic, and is neither easy nor difficult English:

Before long, washed and refreshed, the hobbits were seated at the table, two on each side, while at either end sat Goldberry and the Master.

It was a long and merry meal. Though the hobbits ate, as only famished hobbits can eat, there was no lack. The drink in their drinking-bowls seemed to be clear cold water, yet it went to their hearts like wine and set free their voices. The guests became suddenly aware that they were singing merrily, as if it was easier and more natural than talking.

At last Tom and Goldberry rose and cleared the table swiftly. The guests were commanded to sit quiet, and were set in chairs, each with a footstool to his tired feet. There was a fire in the wide hearth before them, and it was burning with a sweet smell, as if it were built of apple-wood. When everything was set in order, all the lights in the room were put out, except one lamp and a pair of candles at each end of the chimney-shelf. Then Goldberry came and stood before them, holding a candle; and she wished them each a good night and deep sleep. 

The word order "set free their voices" is slightly literary or poetic. The subject matter is a world like that of fairy tales, so there are references to old-fashioned things like "drinking-bowls" and candles and so on.

It certainly isn't the language of Hemingway or Mark Twain!

Having been written by a Professor of English at Oxford LOTR is a masterpiece of English Literature.

"The Ride of the Rohirrim" is probably the best chapter of English ever written.

It took him so long to write but every sentence is a masterpiece.

 "If I may once more refer to my work. The Lord of the Rings, in evidence: the names of persons and places in this story were mainly composed on patterns deliberately modelled on those of Welsh (closely similar but not identical). This element in the tale has given perhaps more pleasure to more readers than anything else in it."

JRR Tolkein