Extra info at the start of a sentence

Your first example is referred to as a Participle Phrase, I've found a fairly simple explanation here. The most pertinent part of the linked article to your specific question seems to be this explanation:

a reason for the action in the main clause:

• Having nothing left to do, Paula went home. (Since Paula had nothing left to do, she went home.)
• Knowing a little Russian, I had no difficulty making myself understood. (As I knew a little Russian, I had no difficulty making myself understood.)
• Working as a sales rep, I get to travel a lot. (I travel a lot because I work as a sales rep.)

Your second example is an instance of a Prepositional Phrase. Since during is not a verb which has been modified, but is a preposition defining a point in the course of time.

This article does an alright job explaining the difference between the two.

Those are adverbial phrases.

In linguistics, an adverbial phrase is a multi-word expression operating adverbially: its syntactic function is to modify other expressions, including verbs, adjectives, adverbs, adverbials, and sentences.

Example
Next year, I will go on the cruise.

— Wikipedia

In this case, the phrases are modifying verbs. You can confirm they are adverbials by asking "the adverb questions":

Particularly, an adverbial at the start of the sentence is called a fronted adverbial.

I will lose weight next year.
Next year, I will lose weight.