*"They had already ten children" is ungrammatical because an adverb, in general, cannot intervene between a verb and a direct object.
The sentence below, however, is OK:
"They have already had ten children"
That's because the first "have" is an auxiliary, not a lexical verb. An auxiliary does not take a direct object.
You need to rephrase. This is a case where you are trying to pack several bits of information into one noun phrase, and it is getting confusing. The solution is usually to split things up:
Here you go:
Population is mostly concentrated in the east. The rest of the region is mountainous, with only a few villages.
Depends what the adjective is describing. Is it describing the entire [noun], or is it only describing the rest of the [noun]?
There are only few villages in the mountainous rest of the region.
Here, you are talking specifically about the rest of the region that is mountainous. The east isn't necessarily mountainous, and your phrasing certainly implies ...
Although you don't want rephrasing, I think you have to. Yes, you would put the modifier in between "the" and "rest"; however, you should rework the second sentence. Try "There are only a few villages in the mountainous part of the region."
The usual place of the adverb in the first sentence is directly before "be". You can move it if you want to; though it may sound odd, it's not impossible.
Your second example sounds a little sarcastic. You could move the adverb to the end to reinforce that meaning:
"There must be order in the class room, at least sometimes!"
Here's a link to a similar ...
No, I would say it is not natural.
The natural position is
That's where I usually go after work.
Usually that's where I go after work
puts emphasis on "Usually", and leaves me expecting something like "but last night I went to the other one".
Its correct, but as Kate says "awkward".
The problem seems to be one of "end weight". Fluent English sentences tend to put longer structures at the end.
Give John the book.
This follows this principle. The two word structure "the book" comes last.
Give it to John.
This also works, the "weightier" prepositional phrase is placed last
The "possessive" is not the reason. The same would be true if you said "the efforts"
The words "all" and "both" are not adjectives. They are functioning as determiners. The possessive "my" is also a determiner. The "main determiner" is "my", and "all" or "both" act as a pre-determiner, and must go before the main determiner:
All the dogs
Both my ...
The negative subjunctive is formed by putting not before the verb, at least nowadays, as indicated in e.g. this Q&A on our sister site English Language & Usage: What's the correct form of the negative subjunctive?. (That seems to imply it used to be the other way around.)
That means the sentence should be
The Senate has decided that such students ...