Translation:The children in the garden are cheerful.
I guess probably depending on context happy, merry, jolly, blithe and all your suggestions should fit.
I assume while pretty, beautiful and gorgeous are rankable, these might be even more blurry.
boldog for happy or élénk for lively are possibly better, but like in English there seem to be a lot of synonyms, which probably all have slightly different meanings. https://szinonimaszotar.hu/keres/vidám
No, I'm pretty sure it can't. For "cheerful kids" you'd need the adjective to come before the noun. And the adjective would be in singular form, because adjectives before a noun don't have to agree in number. Also, if you're saying that the children are in the garden, then you'd need "vannak" in the sentence, because you're talking about where they are.
A vidám gyerekek a kertben vannak" - The cheerful children are in the garden.