Use of preposition у in questions about the magnitude of a quantity

Question

Today, Duolingo taught me that it's Какая глубина этого озера?, but Какая ширина у этой улицы?. My questions are

1. Is that a case of binary grammaticality? I. e., would Какая глубина у этого озера? or Какая ширина этой улицы? be ungrammatical or unusual?
2. If yes, is either of these cases an exception? That is, do/don't you usually use у with physical quantities? If no, is there a rule or can you give me a table relating common quantities to their use of у?
3. What about more "technical" quantities, such as density, potential, or flux?

Answer 1

1. No, either way is correct (in either case), and I even struggle to say which one is more common.
2. I'd say that usually we don't use y with physical quantities (or qualities); plain Genitive is the 'default' way applicable to everything. Площадь кухни, плотность свинца, возраст дочери, напряжение обмотки, цвет стены. So the basic rule is simple: if in doubt, don't use y.
3. As above, generally without y. Having the preposition makes the phrase slightly less formal (or so it feels to me), and technical/formal writing avoids it.

Answer 2

1. Yes, in combination with the question word какой(-ая) in your examples, omitting the preposition у sounds a bit unusual if not awkward. Probably, it's because the question is about assigning a value to an object (like, what depth/width value it has). You don't need a preposition with another question word, каков(а), since the question would focus on the value only (how much its width value is):

Какова ширина этой улицы?
Какова глубина этого озера?

2. A possible context with the question word какая but without need for у (as above, value is in focus):

Какая глубина этого озера сделала бы его судоходным?
Какая ширина этой улицы могла бы исключить пробки?

3. In formal cases (scientific or engineering writings) preposition у is avoided (except when it means near), so the question word is selected so that it provides focusing on value:

Какова скорость света в полипропилене?
Какая температура плавления этого сплава обеспечит заданные характеристики объекта?