How do you call those places where a railroad crosses an automobile
Of course, I've heard how they are called in English, but I suspect
that they are called differently depending on whether the speaker is
from UK, USA, Canada or Australia. So, please, specify in your answer
what type of English you mean.
Not just where the speaker is from. It also depends on which form of English you learn and, which form you use.
First of all, you don't need to use the word automobile with the word road. It is not necessary. Also, the word is not necessary anyway, because it is obsolete and not normally used in the UK.
To answer your first question, those places are called level crossings. The word railroad does not exist in British English. Nobody uses it in the UK. The word railway is the normal British word.
For level crossing:
For the word railway:
I also heard that these places can be called differently even within
one country because of some subtle differences pertaining to those
places. I don't really know what those differences are. It may be the
width of the automobile road, or the difference in height between the
level of the railroad and the automobile road, or, perhaps, the
presence/absence of that red-white thing that goes down and up every
time when the train passes.
Also, how do you call that very red-white thing itself? Especially,
how would you describe its motions in this kind of sentence:
"At the (place), when the (thing) finally (lowered), it was obvious I
would be late for the party"
What words you would substitute for those used in parentheses above?
That “red-white thing” is usually called a barrier. Considering that there are usually two of them, the plural form barriers is usually used. Sometimes they are called gates.
For your sentence, in the UK that can be:
At the level crossing, when the barriers finally came down, it was obvious I would be late for the party.