Th first 4example sentence:
Can you give the length of width of the box besides the height?
Is, as far as I can tell, grammatically valid. I cannot point to any rule or principle which it violates. Yet I must agree with those who told the OP that it did not sound natural, and that any of
- Can you give the length of width of the box as well as the height?
- Can you give the length of width of the box in addition to the height?
- Please you give the length of width of the box and
are significantly more natural, and seem proper to me.
I can only set this down as one of the oddities of English usage. As far as I can understand,. it follows no particular rule, adn simply must be learned and remembered. If ther is a rule I don't consciously know it.
I would not say that "besides" and "as well as" have exactly the same meaning, but the is surely very little difference.
Cambeidge defines "besides" as an adverb or preposition, and gives "in addition to" or "also" as the meaning with examples:
- She won't mind if you're late - besides, it's hardly your fault.
- Do you play any other sports besides basketball?
Cambridge says of "as well as"
As well as is a multi-word preposition which means ‘in addition to’
- She has invited Jill as well as Kate.
- When they go to Austria, they like walking as well as skiing.
The major difference seems to be that "as well as" is not used adverbially. But that is not the use in the example anyway. This gives not bases for the difference between the two example sentences in the question.