Alex+

“And we can’t say: “His room is twice as small as my room.” Instead of this we say: “His room is half as large as my room.”
”

Yes, or you could say "His room is half the size of mine".

In informal speech and writing, native speakers often use expressions such as "twice as small" to mean "half as large", so in that sense it's not true that you "can't say it". However, even though such expressions are grammatically correct, they are logically dubious. There is no numerical measurement of "smallness" such that the first room is numerically twice the measurement of the second. These expressions are therefore best avoided if you want to speak and write precisely.

Alex+

“Road A is three times longer than road B. = Road A is thee times as long as road B.”

Use "three times as long as".

[I clarified this paragraph shortly after posting.] Expressions such as "three times longer" are very common but, again, are logically and mathematically dubious/ambiguous. When you say "Road A is [...] longer than road B", the omitted part is normally a measurement of the difference; in this case, "Road A is 200 km longer than road B". However, 200 km is not "three times" any relevant number, so it doesn't really make sense to substitute "three times".

Alex+

“Jack has five times more apples than Ann. = Jack has five times as many apples as Ann.”

For the same reasons, use "five times as many". 

Alex+

““Ann has five times fewer apples than Jack.” or “Ann has five times as few apples as Jack.””

These are logically dubious in the same way as "twice as small" (and they sound slightly unnatural too, especially the second). I'd avoid these and instead say "Ann has one-fifth as many apples as Jack". Of course, unless you have a particular reason for wanting to mention Ann first, it might be easier to just say "Jack has five times as many apples as Ann".

Alex+

“I’m slightly confused because in my language it’s possible to say :”Road A is three times longer than road B.” or “Road B is three times shorter than road A.” That’s why it is difficult for me to understand than in English we can say “Road B is shorter than road A.” but we can’t say how many times”

English speakers have the same difficulty, which is why these forms of wording are so common. Everyone understands that "three times longer than" is used to mean "three times as long as" and "three times shorter than" is used to mean "one-third as long as" -- and most people go their whole lives without being troubled by the fact that logically it might not make sense. Only a minority of people care.

Alex+

“Q: Which is more common?
“His room is half the size of mine.” or “His room is half as large as mine.””

Probably the former. Also common is "His room is half as big as mine."

It is not an egregious error in English to say "three times shorter than". If you do, then everyone will understand what you mean and many people will not register it as an error at all, especially in conversation. In formal writing I would avoid it.