Translation:Even though it is not cold, she wears a coat.
It is clipped. I live here and the recordings are a little shoddy, especially for some of the articles s/a 'het'. Also, in my opinion, she kind of has a Belgian accent (but that could also just be me). I'm hoping with time and more user contributions it will get better :)
The inversion of which you speak only occurs in the interogative (questions) and in the second person. for example: "Jij doet"(you do) vs. waat doe je? (what are you doing?). Even though there is inversion of subject and verb in the exercise, it does not require the use of the first person form of the verb.
"Hoewel het niet koud is" is a subordinate clause. The verb in a subordinate clause always goes to the end. "draagt zij een jas" is the main clause here.
It would be "Zij draagt een jas" if the main clause came first. However, in this case the subordinate clause comes first and the main clause comes second.
When a sentence starts with a subordinate clause, as in the second example above, the very first word after the comma (before the main clause) MUST be the verb. That's why it is "draagt zij een jas" in this case.
There is no second sentence; there is only one sentence here. It's what's known as the 'main clause' of the sentence. If the main clause follows a subordinate clause or an adverbial phrase, the verb and the subject in the main clause get inverted. So if the main clause comes before the subordinate clause/adverbial phrase, then the subject and the verb are not inversed.
The same rule applies to English as well: Under the bed hides the mouse or Up the tree climbs Bob. It's a rather literary feature nowadays and is uncommon in spoken English. I haven't done any research but I suspect this feature might've been in more common use a few centuries ago.
Your English sentences aren't the same grammatically though, as they are only one main clause. What you've done is start the sentence with a prepositional phrase, not a subordinate clause. There is no verb in "up the tree" or in "under the bed", which is why these are not subordinate clauses -- both are just a preposition + its object.
That is indeed a very astute observation. V2 inversion in declarative sentences are fairly limited in English usage nowadays. Most of the use cases nowadays centre around prepositional phrases (as you mentioned) and conditional sentences ("Had he not turned up in time, she would have died."). However, this feature was a staple in Old English, from which English is derived (at least as far as grammar is concerned). Whatever exists in English now, is simply a remnant of that. So essentially I was trying to demonstrate this inversion feature through whatever example people may be familiar with in English rather than trying to find a direct mapping, which is not possible really.