Learn Russian: Нам страшно в этом жутком заброшенном месте. - We are scared in this eerie abandoned place.

In Russian, subjective feelings and physical states are often expressed using impersonal constructions. The person experiencing the feeling goes in the Dative case (нам), and the state itself is an adverb-like word (страшно). If you said мы страшные, it would literally mean "we are scary" or "we are ugly"!

Could I use the verb бояться (to be afraid) here instead of страшно?

Yes, you could say Мы боимся в этом... месте. However, нам страшно emphasizes the spontaneous feeling of fear or creepiness induced by the environment, whereas мы боимся sounds slightly more like an active, ongoing state of fearing something specific. In a spooky place, нам страшно (it is scary to us) is a very natural choice.

What case is в этом жутком заброшенном месте, and why?

It is in the Prepositional case. The preposition в (in) takes the Prepositional case when describing a static location (answering the question "where?"). Since место (place) is a neuter noun, its modifiers take the neuter Prepositional endings: этом, жутком, and заброшенном.

What type of word is заброшенном and how is it formed?