"I will never ever leave you!"
Translation:Jag kommer aldrig någonsin att lämna dig!
ska works too, the difference between kommer att and ska in a sentence like this is that ska is very clearly a promise. kommer att can of course also imply a promise, and often does, but it doesn't necessarily mean that, whereas ska does.
As berniebud says, tänker is really closer to 'intend to' in meaning. (but all three are accepted answers).
The strange thing is that the adverbial aldrig någonsin goes between the two parts of kommer att. But as HelenCarlsson pointed out in a previous comment, you would do the same in English in a construction like am going to – you would say I am never ever going to … and the adverbial would go between 'am' and 'going to'. So while it doesn't work exactly the same, it's pretty similar.
I guess you know that a sentence adverbial splits two verbs (in a main clause), e.g. "jag ska lämna dig" and "jag ska aldrig lämna dig", but the tricky thing here is that the sentence adverbial even comes before "att" in "kommer att", right?
Think of "I am never going to leave you" vs "I am going to never leave you". The sentence is about something you will never and not something you will do never. I hope this was not too confusing :).
I just want to expand on your second paragrah and provide the translations in case that makes it easier for the op.
Jag kommer att aldrig lämna dig. = I am going to never leave you
Jag kommer aldrig att lämna dig = I am never going to leave you.
If you're a native english speaker (I'm not sure if you are), you can see that the first english sentence, "I am going to never leave you" kind of makes sense, but it doesn't really convey exactly what you're trying to say, and isn't the best way to say it. That's similar to how it is in the first Swedish sentence.
The second sentence obviously sounds a lot more idiomatic, and is something you would expect someone to say, and the same goes for the second Swedish sentence.
While I am going to never leave you could work in English, Jag kommer att aldrig lämna dig is just ungrammatical to my ears in Swedish. I think if you ever see that from native speakers, it's just from people who don't normally use att in kommer att, so they have no feeling for where to put att. What do you think, Helen, does it sound OK to you?
I think jarrettph:s description 'taking the action of not leaving them rather than not taking the action of leaving them' is spot on – and as of now, as far as I can tell, this works in English but not really in Swedish. (To be totally honest, you might sometimes hear this kind of thing in Swedish, but it's an anglicism.)
The English one sounds pretty OK to me, not that that really means a lot since it isn't my native language (and maybe I misread AlecHirsch1s comment), but I was thinking of other things they can say too, like 'Could we not talk about this?' where the negation doesn't really work the same way in Swedish. (Though we might be in the process of importing that 'emphatic not' from English.)
I found the expression here http://blogs.transparent.com/swedish/swedish-sentence-adverbials/. It seems that it does not exist in English though (as you said), so it's probably just a translation of the Swedish expression "satsadverb" (or "satsadverbial").
Different terms are needed for different languages. This term is important when teaching Swedish grammar because a sentence that modifies the whole sentence will go in a special place in the sentence, as opposed to other kinds of adverbials. Also see my longer post about word order: https://www.duolingo.com/comment/8970470
att lämna is really one word, the infinitive, "to leave". Att does not go with kommer so the adverbs go before att lämna. English is fairly calm about split infinitives like "to boldy go". Many European languages have just a single word for the infinitive eg aimer in French is "to love" so it is impossible to split the word in two.