Translation:The children are playing there, where the grass is nice and long.
Hosszú isn't an adverb. Granted, the translation might have changed after you have written this, but both szép and hosszú are adjectives. It might be that hosszú, however, functions as a noun here, which the adjective szép modifies, so you're left with something like "the grass is a nice (long one)". In any case, the "szép hosszú" construction is a quite common one.
Hungarian adverbs are usually(!) made by adding -an/-en to the adjective: szépen (you should know that one already) and hosszan.
Also, for your translation "where the beautiful, long grass is", you'd need to restructure the clause a bit: "ahol a szép, hosszú fű van."
No, you cannot use van here. In sentences like "[noun] is [noun]" or "[noun] is [adjective]" (so-called copula sentences) you never use van.
A kutya az ember legjobb barátja - The dog is the man's best friend.
My idea to use hosszú as a noun here might be a bit off, I'm in a slight dispute with myself here. Anyway, this construction of "szép [adjective]" does not exist in English, but it does in German, like I've realised: "schön [adjective]". I would just take it as a special grammatical contruction. The szép enhances the following adjective: "Szép puha, szép zöld, szép hosszú a fű"
Since this construction doe not exist in English, you can't translate it too literally, but "nice and [adjective]" should do the job here.
Lhorton, it doesn't really match the Hungarian sentence, though. The second clause of the Hungarian sentence describes what the grass is like at a specific spot (suggesting that there's grass all around, but it's mostly shorter), but you're saying that there is grass at a specific spot. Translated back into Hungarian that would be "... ahol szép hosszú fű van."
That translation is (was) kinda weird. Szép is an adjective (the adverb would be szépen) and I suspect that hosszú acts as a noun here. English has a bit of trouble with adjectives as nouns, so instead of an adjective-noun "the grass is a nice (long one)", it prefers adverb-adjective constructions: "the grass is nicely long."
I think all the cases should be accepted.
For the English words?
"Beautiful" is mainly a physical thing - in its most basic meaning it means that the look of something is pleasing to you. In a more metaphorical sense it can also refer to nonphysical traits - a sound or a person's mindset. It all has the meaning that it's aesthetically pleasing to you. Exposure to beautiful things will inevitably make you happy.
"Nice" can also be used for physical things, but it's somewhat weaker than "beautiful". It doesn't have to do much with (objective) aesthetics, but generally giving you a pleasing feeling, usually somewhat more diffuse, you don't exactly know where it's coming from. Describing a person as "nice" is different, though. That means that the person is kind or friendly.
The meaning is correct, but the issue is that "pretty long" usually means something else in English: "quite long" or "very long". Using "pretty" as an adverb gives the following adjective an increased intensity, but doesn't say anything about the actual prettiness.
"Isn't it pretty?"
"Pretty dull, yeah."