Learn Russian: Защитив диссертацию, учёный получил степень. - Having defended the dissertation, the scientist received a degree.

Защитив comes from the perfective verb защитить (to defend). To form it, you take the masculine past tense form (защитил), drop the -л, and add -в. Because it is perfective, it tells us this action was fully completed before the main verb: first he defended it, then he received the degree.

Why does диссертацию take the accusative case here if there is no regular verb in that part of the sentence?

Even though защитив is a verbal adverb, it still takes the same objects as its base verb. Since защитить takes a direct object in the accusative case, защитив does exactly the same.

The word учёный has an adjective ending. Why is it acting as the noun subject?

Учёный is a substantivized adjective. This means it originally described a noun (a 'learned' person) but is now used purely as a noun meaning 'scientist'. It functions as the subject here, though it still declines like a regular adjective in other cases.

What gender and case is the word степень in this sentence?

It is a feminine noun in the accusative case, acting as the direct object of получил (he received). Because степень is a feminine noun ending in a soft sign, its accusative form looks exactly the same as its nominative dictionary form.