Volgens de burgemeester verbetert de nieuwe tramlijn de stad.

Volgens is a preposition meaning according to. It introduces the source of information or opinion. In Dutch you follow it directly by a noun or pronoun (no extra words in between). Here volgens de burgemeester means according to the mayor.

Dutch main clauses generally follow a Verb Second rule: the finite verb is always in the second position. When you start the sentence with any element other than the subject (here: volgens de burgemeester), you treat that as position 1. The verb (verbetert) then moves to position 2, pushing the subject (de nieuwe tramlijn) to position 3.

Using de indicates you’re talking about a specific mayor, presumably the one everyone knows about. If you said een burgemeester, it would sound like any mayor in general, which doesn’t fit when you want to cite a particular person’s view.

In Dutch, attributive adjectives (adjectives placed before a noun) take an -e when the noun has a definite article (de/het) or is plural. Here:
• de nieuwe tramlijn
• definite article (de) + common gender noun (tramlijn) → adjective gets -e.

The direct object is de stad (the city). The verb verbeteren is transitive, meaning it needs an object that receives the action. The tram line doesn’t improve itself; it improves the city.

Yes. You could say:
De nieuwe tramlijn verbetert de stad volgens de burgemeester.
In that version the subject (de nieuwe tramlijn) stays in first position, so you don’t invert verb and subject there.

Yes. Verbetert is the 3rd person singular present form of verbeteren (to improve). The subject de nieuwe tramlijn is singular, so you add -t.