is it possible to use "mnoho" here?
Yep, I would actually prefer it in written Czech. 'Hodně' is more colloquial.
Why does the location clause always have to stand at or near the beginning of the sentence (of course after the 2nd position items)? Or is it a case of “it sounds weird” if it doesn’t?
It does not have to. At least not in Czech.
Anyway, you should always report the sentence that was rejected, if you had such a sentence.
I will. But is it the “preferred” word order?
Which one? This really can't be answered.
Here probably the more natural word orders happen to begin "V Česku" but that is because of the meaning, it is strange to stress "in Czechia" by placing it at the end. The grammar absolutely does not prohibit placing it there.