"The professor's computer is expensive."
Translation:حاسوب اَلْأُسْتاذة غالي.
Yep, that would be incorrect.
Think of it that way: حاسوب and أستاذة are a compound (2 nouns added to each other to give a meaning). In such compounds, the definite article goes to the second noun in the compound. The addition of "AL" is pretty important here because it is the marker that tells that the first noun is added to the second noun (i.e. "AL" is the chain the links the two nouns). Hence: حاسوب الأستاذة (computer of the she-professor).
yes, but adjectives come after the noun not before the noun.
Also, when the adjective is attributive and the noun is defined, then this adjective must be defined (with AL).
In the sentence above we have the adjective غالي (expensive), but here it is not defined with (AL) even though الأستاذة is defined. This is because this adjective is telling information (called predicative) and not attached to the noun (not attributive). You can see that in English by the fact that (expensive) comes AFTER the verb (is); This is how English deals with predicative adjectives.