That's a good rule of thumb and it's mostly accurate but it should be borne in mind that there are certainly exceptions. For example, Glücks is nowadays considerably more common than Glückes (despite being a one-syllable word), and Gesundheitszustandes is considerably more common than Gesundheitszustands (despite being a five-syllable word) - the reason for this is not entirely clear, but it probably has to do with how well the word "flows"
An interesting fact is that, in German, adding a syllable to mark declension (outside plural formation of course) is in a slow decline. In Old High German, the dative singular took an -e as a rule, and the genitive -es, this stayed the same in Middle High German but it was now common to use the short form with endings like -el or -er: des Zimmers and dem Zimmer. Interestingly, in New High German (what is spoken today), there is a tendency to use the short form where it is unnecessary and the short form is now overall more common, Beitrags instead of Beitrages, Films instead of Filmes, Königs instead of Königes (Königes actually sounds poetic), but this hasn't caught on with many simple words like Landes or Kindes. Moreover, marking the dative singular with -e has been declining for centuries and finally became archaic around the second world war (it had disappeared from the spoken language long before, but remained present in literature); although it still exists to this day in parts of rural Saxony/Thuringia (rapidy disappearing).
I am a native English speaker (American) and I would most certainly say that. Indeed, the German sentence says "das Haus," which to me means that the English sentence should say "the house." If you have to say "the house," then "the house of this man" is a perfectly reasonable construction. Unfortunately, Duo now finds that to be incorrect, and insists on "That is this man's house," which to me seems like a less faithful translation of the German.
The genitive attribute is usually placed after the possessum (not necessarily nominative), except with names:
- Peters Hund
- Deutschlands Wälder (also: die Wälder Deutschlands)
Otherwise it sounds old-fashioned or archaic:
- meines Vaters Arbeit (old-fashioned, modern: die Arbeit meines Vaters)
- bis an der Welt Ende (archaic, modern: bis ans Ende der Welt)
You just have to go off of context. It's the same with 'Frau'. It could be lady/woman, or it could be wife. In this particular sentence, there isn't anything that implies that we're specifically talking about a husband, so you would use a more general definition of "Mann"..
If you wanted to be more specific, you could use der Ehemann or der Gatte (Husband) or die Ehefrau or die Gattin (Wife).
To add to that, I knew someone from around Braunschweig (North Germany) who used the genitive quite freely and rather often in everyday language, which is a major contrast to southern dialects as you said.
Also, the genitive is non-existent in almost all dialects of German (the only major exception is Walliserdeutsch in the very south of Switzerland). Dialectal varieties are not the same as colloquial varieties of standard German, in which the genitive is used to a limited extent, depending on region. The reason why the genitive is so rarely used in colloquial language in Austria and Southern Germany is because most people there still speak the genitive-free dialect, not a colloquial variety of the standard.