I'm pretty confused about how to use the e ending. It's used for -ly so li ludas ĝoje is correct. But what about for example li estas vive? Is that acceptable? Basically how does the e ending work in terms of "estas"?

The -e ending doesn't necessarily correspond to -ly in English. While -a, -aj, -an, -ajn denote adjectives, -e and -en denote adverbs. Many adverbs in English do end in -ly, but not all (e.g. well); on the other hand there are words that end in -ly, but are not adverbs (e.g. lively is an adjective).

In Esperanto we use adverbs to describe verbs (1), adjectives (2), adverbs (3), clauses (4) and something unsaid, omitted or inferred from context (5).

(1) Mi kuras rapide. = I run fast. (rapide describes kuras) -> Same as in English
(1) Kuri estas facile. = Running (lit: to run) is easy. (facile describes kuri) -> Unlike English

(2) Ĝi estas sufiĉe granda. = It is big enough. (sufiĉe describes granda) -> Same as in English

(3) Li kuras ekstreme rapide. = He runs extremely fast. (ekstreme describes rapide) -> Same as in English

(4) Ne estas saĝe, ke vi enlitiĝas tiel malfrue. = It isn't wise that you go to bed so late. (saĝe describes ke vi enlitiĝas tiel malfrue) -> Unlike English

(5) Bone. = OK. (bone describes the whole situation) -> Depends

But what about for example li estas vive? Is that acceptable? Basically how does the e ending work in terms of "estas"?

"Li estas vive" means "He exists in the manner of something alive/living" (credit to RiotNrrd). "Li estas viva" means "He is alive". "Li estas verva" means "He is lively." Remember, that "lively" is an adjective although it might look like an adverb.

In clauses of the type "A estas B", the following rules apply:

If A is a singular noun, B ends in -a: La kato estas dika.

If A is a plural noun, B ends in -aj: La katoj estas dikaj.

If A is anything else, B ends in -e: Kuri estas facile. Ke vi estas juna, estas vere. Sufiĉe estas sufiĉe.

There are some other fringe cases, but for 99% of all sentences the above rules will hold.

The following might go beyond the scope of your question, but let me mention it just for the sake of completeness.

1. Some adverbs in Esperanto don't have any ending at all.

1a. These include the correlatives in -am (which is not an ending, but part of the root): iam, kiam, ĉiam, tiam, neniam. Those are adverbs of time: at some time, when, always, then, never.

1b. These include the correlatives in -om (which is not an ending, but part of the root): iom, kiom, ĉiom, tiom, neniom. Those are adverbs of measure: some, how much/many, all, that much/many, none

1c. These include the correlatives in -al (which is not an ending, but part of the root): ial, kial, ĉial, tial, nenial. Those are adverbs of reason/cause: for some reason, why, for all reasons, for that reason, for no reason

1d. These include the correlatives in -el (which is not an ending, but part of the root): iel, kiel, ĉiel, tiel, neniel. Those are adverbs of manner: in some way, how, in all ways, in such a way / so, in no way

1e. These include the correlatives in -e (which is not an ending, but part of the root): ie, kie, ĉie, tie, nenie. Those are adverbs of place (locatives): somewhere, where, everywhere, there, nowhere.

1f. These include the words in -aŭ (which is not an ending, but part of the root): e.g. morgaŭ, baldaŭ = tomorrow, soon

1g. These include some other words: e.g. nun, jam = now, already

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2. You can form adverbs of place (locatives) from nouns by using the ending -e: hejme, lerneje, arbare = at home, at school, in the forest.

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3. You can form adverbs of direction (allatives) by using the ending -e combined with -n.

3a. From nouns: hejmen, lernejen, arbaren = home (going home), to school, into the forest

4b. From the words of 1e: ien, kien, ĉien, tien, nenien = to some place, whereto / whither, to all places, to that place / thither, to no place.

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Ree and reen are a bit oddball, but "ree" works like 2 (and like a "normal" adverb) and "reen" like 3.

"Li estas vive" isn't grammatical. "Li estas viva" means "He is alive". "Li estas verva" means "He is lively."

I kind of have to take issue with this. It IS grammatical. We're using the root viv- to describe "estas" instead of "li", but you're allowed to do that. "Verva" means "lively" in an energetic way, the way we mean it in English, but "vive" ALSO means "lively", just in the more muted way of "in the manner of being alive"; "livingly" might be a better (though mangled English) translation.

Li estas verva = he is lively (and energetic, excited, bouncing around, etc.).
Li estas vive = he exists in the manner of something alive. Basically, he isn't dead, OR, perhaps he IS a dead automaton that merely gives the impression of being alive. Context would reveal which was which. I would almost want to write the sentence as "Li estas, vive", but the comma is not actually required.

It's not the sort of thing that probably commonly gets said, but I claim that it is not ungrammatical. Unusual, perhaps. Very unusual, probably. But not against the rules, and not devoid of meaning.

RiotNrrd:"Li estas vive" isn't grammatical. "Li estas viva" means "He is alive". "Li estas verva" means "He is lively."

I kind of have to take issue with this. It IS grammatical.

I concur. It is grammatical. I have corrected my original post.

As you have already pointed out, in all but the most esoteric cases, "He is lively" probably means "Li estas verva."
I didn't even know that "lively" can also be an adverb. But, hey, English isn't my native tongue. I also just learnt that "livelily" exists.

Apparently, Old English made a distinction between e.g. adjectival nihtlīċ (nokta) and adverbial ‎nihtlīċe (nokte), but now in English this distinction is lost and it is just -ly.

nornen:

Thank you this helped me a lot. I hadn't considered that there were so many scenarios that -e works in. I'll probably need to study this more but this definitely clears up a lot of things.