I'll focus on your questions here. First, "more easily" is correct. Second, using "more" and "the most" is the correct way to do comparatives and superlatives with adverbs. Here are some examples:
This is easily done.
This is more easily done.
This is the most easily done.
You can also put the "done" before the adverb as you have it. For the most part the word order is a matter of style, and perhaps also what you want to emphasize.
Note that we usually say "the most easily" here rather than "most easily." If you just say "This is most easily done" you are actually saying "This is very easily done" with a little more emphasis than "very". You've probably heard the phrase "That is most interesting" for example. It doesn't mean that that concept is the most interesting of several concepts; it simply means that it is very (extremely, highly) interesting. On the other hand, if there is enough context to imply the superlative meaning, the "the" can be left out. I'll borrow from 200_success's example here:
Calculating this antiderivative is fairly easily done using integration by parts, but it is (the) most easily done using a trigonometric substitution.
As for -er and -est vs. more and most: we use the -er and -est suffixes on shorter adjectives. On longer ones, we use more and most: you are most unlikely (see how "most unlikely" works?) to see "sympatheticer" or "sympatheticest" anywhere, for example. We always use more and most with adverbs (at least, I can't think of any offhand that use the -er and -est suffixes).
Now, with all that said, this is a rule that is often broken by using adjective-adverb subsitution, especially in conversation. For example:
Joe got there quicker than Paul.
My car runs slower than his.
You will often see this sort of thing. Two reasons come to mind. First, adjective-adverb substitution happens regularly, not just in comparatives and superlatives. "My car runs slow (fast, good, bad)" is often substituted for "my car runs slowly", for example. Second, it's a more economical way of expressing the idea. You might well see this, especially if the writer is adopting a conversational or narrative tone:
It seemed that Joe was running as quickly as he could across the field, but when the bull started taking notice, he ran quicker. A lot quicker.
You'll see that the sentence uses the adverb in the regular form, and then switches to the adjective for the comparative.