Learn English – Is it correct to say “which country do you like the most?”

Most has two closely-related meanings.

1 the largest in number or amount
2 more than half / almost all of somebody/something

As that OALD link says, the most is often used for the first meaning - but informally, the is often omitted. It's never used with the second meaning.

Suppose The Queen of Hearts were to ask "Who ate my tarts?"...

A1: "The knave ate the most"
A2: "The knave ate most"

If several courtiers ate one tart each, but the knave ate two, then either answer could validly be given, since he ate more than anyone else (but note that A1 is a "very slightly formal" usage).

But if the knave actually ate more than half (but not all) of the tarts, only A2 is correct. Though the Queen might not understand that intended distinction, since the could be omitted anyway.

It's worth making the point that the is never included when most simply means very...

"I am most grateful for your help"

...but interestingly, whereas in...

"There were 50 people there, at [the] most"

... the is normally omitted, it's almost always included if we add very as an intensifier...

"There were 50 people there, at the very most"

Both of your examples

This is done easily.

This is done more easily than I thought.

are grammatically correct.

However, I would personally consider both sentences to be slightly awkward or incomplete, since you could express the same ideas using adjectives. If you had no more details to add, then more natural expressions might be

This is easy.

This task is easy.

This was easier than I thought.

This job was easier than I thought.

The original examples would also be fine if there were more details to complete the sentence, so that there is a reason to use the adverbial form rather than the simpler adjectives that I proposed above.

Calculating this antiderivative is easily done using integration by parts.

Calculating this antiderivative is more easily done using a trigonometric substitution.