Learn Hardcore Dutch: Het regent hard; de tiener wil echter nog steeds zonder regenjas naar buiten gaan. - It is raining hard; however, the teenager still wants to go outside without a raincoat.

In Dutch weather expressions, het acts as a dummy subject (an impersonal pronoun) just like it’s in English it’s raining. You always say het regent, het sneeuwt, etc., even though het doesn’t refer to any specific thing.

What part of speech is echter, and why is it placed after the verb in de tiener wil echter?

Echter is a conjunctive adverb meaning however. In Dutch main clauses, adverbs of this type normally follow the finite verb, so you get Subject – Finite Verb – echter – rest of clause.

Why is nog steeds used here, and how does it differ from just nog or steeds?

Nog steeds literally means still (continuing up to now).

Nog alone can sometimes mean still, but is more ambiguous.
Steeds means always or continually without the nuance of up to now.
Using nog steeds emphasizes that the behaviour truly persists despite the rain.

In zonder regenjas naar buiten gaan, why is there no article before regenjas?

After the preposition zonder (“without”), you can omit the article to express a general lack of that item. If you want to be more specific you could say zonder een regenjas, but omitting the article is common to mean simply without a raincoat.

Why does gaan appear at the end of naar buiten gaan instead of next to wil?

Wil is a modal verb and occupies the second position in the clause. Its full infinitive gaan remains at the end together with its complement naar buiten. Hence you get wil ... naar buiten gaan.

What is the function of the semicolon (;) between the two clauses?

A semicolon links two closely related independent clauses without using a conjunction. Here it highlights the contrast between It’s raining hard and the teen still wants to go out without a raincoat. You could also use maar (“but”) with a comma:
Het regent hard, maar de tiener wil nog steeds zonder regenjas naar buiten gaan.