Because the focus of the sentence is "oda" (it has to be as you are contrasting with "ide"). The focus must go directly in front of the verb - akarok. (Forget all that "free word order" stuff - you have some choice but often you are tightly constrained.)

Me too! But I can tell by your incredibly sentence that you are doing better than I am... Have a lingot to encourage you!

The verb "kér" does not mean "want", but rather "ask" or "request". It's inappropriate here, and I don't think it can be used at all with a verb phrase, only with nouns.

(You may be mixing these meanings up because you can order food as "pizzát kérek", which can informally be translated "I want pizza" or "I would like some pizza", but a direct translation there would also be "I request pizza" or "I ask for pizza".)

"Kérek" can only be used to translate "I want" when you could also say "I would like to have." It means "I am asking for" and can only be used to request objects.

• Kérek egy gombóc sztracsatellát. = "I want one scoop of stracciatella."

report it as a solution that should be accepted

No. "Itt" and "ott" simply refer to a place, while "ide" and "oda" include motion too.

• "Ott akarok lenni" -- "I want to be there" (in the park)
• "Oda akarok menni" -- "I want to go there" (to the park)

"Here" in this sentence includes motion just like "there", it's just referring to a different place.

"Én akarok odamenni." (note the explicit Én) basically emphasises that "it is me, who wants to go there". But for simple statement, the odamenni, needs to split, and becomes "oda akarok menni, nem ide".

Thank you Fiery, this is how I learned as a child. Akarok is a little too demanding

Shamarth's explanation above says why your version is wrong. I don't know anything about the word order or any of that, but since the sentence we were translating was "I want to go there, not here" ott and itt would be incorrect since they are being in those places rather than going to those places.

It has a different emphasis, a different meaning.
"Oda akarok menni" is plainly saying that "I want to go there".
"Oda menni akarok" is saying that "I (REALLY) want to go there (and not check it out via webcam / other people's photos)". The emphasis is on the "to go" and implying that there is another option that you want to disagree with.
And since the sentence is really just a clarification on where I want to go, the first one should be used.

No, that's ungrammatical and misses some essential parts of the sentence to be translated.

You need the verb akarok to say "I want." (There are other verbs to express "want", but you have to use one of them.)

You also need the directional forms ide and oda ("to here" and "to there"), not the static forms itt and ott ("here" and "there").

"vagyok" means "I am" as in "I exist" or "I am described as". In Hungarian it is never used as part of a verb like in English "I am going".

There is no such words as "ida" and "ode". Vowel harmony means itt must take a front vowel - so "ide" and ott must take a back vowel - so "oda".

Strictly "szeretnék" is "I would like" - which could be loosely thought as equivalent to "I want" - although it is much, much more polite and uses a different verb conjugation. It takes an infinitive ie "menni". "menj" is second person imperative ie Go!

Yes. As you are comparing "oda" with "ide", "oda" has to be in focus ie before the verb.