Saluton el Barato!
I have been learning Esperanto since last year but the behaviour of Esperanto adverbs still puzzle me. The use of Esperanto adverbs give me nightmares, for example (Ekzemple):

1. If Ekzemple is "for example", can Enkonduke mean "for Introduction"?

2. If Endkonduke means "with introduction/for introduction" can 'ame' and 'malame' mean 'with/for love and hate' respectivly?

3. What does 'Elkore Saluton' literally mean?
Can it not be replaced by "kore Saluton"?

4. Can 'Cxu vi kuiras Cxine' mean "Can you cook in China, can you cook for China and can you cook by/with China"?

Amike,
Raj.

1. Yes, this appears to be a good usage of the word. Ekzemple means "for example" or (If you'll permit me to violate English grammatical rules) "for example-ly".

2."ame" would be something like "lovely" or "lovingly" while "malame" would be "hateful" or "hatefully".

3. "Elkore saluton" can be roughly translated as "from-the-heart-ly, hello". Kore and elkore are nearly the same, with only a hint of a difference.

4. "Cxu vi kuiras Chine?" means "Can you cook Chinese (style)?".

It looks like the problem you have is turning other parts of speech into an adverb. In English, the best case is when you can use the termination -ly to do this directly:
adj. nice -> adv. nicely
subst. cost -> adv costly

English does not always allow this, and then you need to use an adverbial phrase. But beware, the preposition that helps form the adverbial phrase is not the same for all cases, and is not the same preposition for all languages. For example for "for example" , in German you wouldn't say "für Beispiel", but you'd say "zum Beispiel" using the preposition "zu" which translates to English as "to".

I think that the rule of thumb for Esperanto E words is that the meaning is the one that makes the most sense. When you translate this E word into English, you might need a preposition. But there is no single preposition to use in all those cases.

"Elkore" helps you by containing the preposition: "el" -> "from". "Elkore" means "from the heart". "Kore" means "heartily". Not much difference.

"Cxu vi kuiras cxine?" I would translate as "Do you cook chinese (food)?". The other variants can be translated into EO by using prepositions:
"Do you cook in China?" -> "Cxu vi kuiras en Cxinio?"
"Do you cook for China?" -> "Cxu vi kuiras por Cxinio?"
"Do you cook with China?" -> "Cxu vi kuiras kun Cxinio?"
Although, to be honest, I'd never ask any of these questions

I freely admit that I am not a professional grammarian or (structural) linguist, just a person who has been fascinated by human language for long decades. Original proto-Indo-European had an instrumental case for nouns. A few of its later descendants retain this case, or at least something functionally like it. Languages of some other major families also have an instrumental case or something functionally like it. As I have read many Esperanto texts over time, it seems to me that Esperanto so-called "adverbs" in '-e' may in practice be treated as instrumentals. As far as I recollect, Zamenhof knew Russian, and he may have been thinking in terms of the Russian instrumental case with regard to some so-called adverbs in '-e'. Even as a non-expert, I find it a useful construction.

bartlett22183:As far as I recollect, Zamenhof knew Russian

It was his mother tongue (together with Yiddish).
And yes, the adverb often works like an instrumental, but it is much more than that, in fact the most flexible grammatical class in Esperanto.

bartlett22183:I freely admit that I am not a professional grammarian or (structural) linguist, just a person who has been fascinated by human language for long decades. Original proto-Indo-European had an instrumental case for nouns. A few of its later descendants retain this case, or at least something functionally like it. Languages of some other major families also have an instrumental case or something functionally like it. As I have read many Esperanto texts over time, it seems to me that Esperanto so-called "adverbs" in '-e' may in practice be treated as instrumentals. As far as I recollect, Zamenhof knew Russian, and he may have been thinking in terms of the Russian instrumental case with regard to some so-called adverbs in '-e'. Even as a non-expert, I find it a useful construction.

No, an instrumental is something quite different and not particularly required in Esperanto. For example in Polish we don't have to use a word for Esperanto "per" because this is expressed by adding a special ending changing the meaning of a word - it's similar to the Esperanto -n but has another function.

Example:
ręka - mano
rękę - manon
ręką (ręcznie) - per mano (mane)

Esperanto adverbs are plain adverbs and like adverbs have many functions - decribe time, place and a manner of doing something. As you can see, sometimes they can be also used like an instrumental, but it's only one of their many other functions.

I failed to express myself adequately. I did not mean to imply that all E-o words in '-e' are instrumentals, only that some have the functions of instrumentals in some instances. Yes, in many instances, '-e' words are plain adverbs.