I hear both Т and Д in the recording. And that's how we pronounce it, too. Maybe it could be omitted in fast speech, hmm... But it's definitely there in the recording.
As for your first question. If the second means multiple houses it is домА. But the first means "at home" and that's a different meaning and a different pronunciation: дОма.
Hm, in real-life conversation it would depend on intonation, but here it should depend on the word order, so I'd say it means the first. And that's how I understood that sentence when I saw it, too. If I wanted to write the second it would be this: "Тим дома всегда отдыхает". No negative connotation though, just a fact. "Do nothing but lie around" sounds quite negative =)
I feel like there is normally something slightly negative about this sentence in Russian...still depends on intonation, of course, but it feels like a common complaint about someone who doesn't work hard enough, etc. Let me know if I'm wrong, but I would translate it as "he is always resting at home"
In Russian, word order and semantic stress can influence the meaning of a sentence. You can stress the word "отдыхает" to report what he's doing at home - Дома он отдыхает or Он дома отдыхает. The sentence Он отдыхает ДОМА can be the answer to the question Where does he rest? If you say Дома отдыхает ОН, it means that nobody rests at home except him.
Agreed. As a result I typically have forvo up in a separate tab to run pronunciations through their database. It's a little frustrating, but it works at least. It would just be nice not to have to utilize a third party so much for something as simple as pronunciation.