Posts 2,5,6,10,15 and 17 have given pretty good feedback on the original question of how an AE speaker would understand "box of sweets".
From Keith's description for BE, I have learnt about the recent addition to the definition of the items from India or Pakistan but he does not use the term "box of sweets" for a box containing "sweet dishes served as dessert."
You have found some examples both of a
different definition (
sweet BRIT *a pudding, fruit, or
any sweet dish served as a dessert) and examples of things that Americans refer to as candies. The word sweet is versatile and differs in different locations/neighbourhoods. Many of the "sweets" in the picture in your BBC link are not "sweet dishes served as dessert", but things an AE speaker would expect to find in a candy store, not on a dessert menu. A BE speaker would put them in the confectionery category (it includes fudge, caramels etc). On that site are also items that could fall into the "sweet dish served as dessert" category.
Perhaps you can re-state your question if it remains unanswered")
* Kate Fox (in "Watching the English") has some interesting comments on this particular use of the word "sweet" to mean dessert - it is not used uniformly throughout BE society!