We have three words for a car!!!
!אוֹטוֹ, מְכוֹנִית, רֶכֶב
Is there any essential difference between them? Are there rules about when and how to use each of them?

Well, the good news are - these are synonyms. :) They mean the same and can be used interchangeably. Absolutely. :)
However, as it always happens with synonyms - in any language - there are certain nuances that each synonym conveys, it, sort of, gives a different feeling, different set of associations is evoked when we hear it.

To demonstrate it, the best one to start from is אוטו - it has a very distinct "emotional coloring". When I say אוֹטוֹ I am thinking of אוֹטוֹ גְּלִידָה, I am thinking of toy cars and I am thinking of a song הָאוֹטוֹ שֶׁלָּנוּ גָּדוֹל וְיָרֹק that every Israeli child knows. (Here is the link, if you want to listen to it.)
So, as you've seen, this word doesn't have the connotations of a prestigious vehicle - it's rather a word one would use with kids. :)

*Again, it's not a strict rule - these words are really synonyms, use any in any situation and it won't be a mistake - what I'm saying here is just that each evokes a certain set of associations, that's all.

This is the time to have closer look at the last word in our collection - רֶכֶב.This one is the oldest of them all, perhaps, - it appears already in the Bible - nearly in the same meaning! Remember the line: וַיָּשֻׁבוּ הַמַּיִם וַיְכַסּוּ אֶת הָרֶכֶב וְאֶת הַפָּרָשִׁים לְכֹל חֵיל פַּרְעֹה הַבָּאִים אַחֲרֵיהֶם בַּיָּם לֹא נִשְׁאַר בָּהֶם עַד אֶחָד.
(שמות פרק יד)Back then these were not exactly cars that were thrown into the sea, but the most common transport of those times, similar to today's cars - chariots. :)

Another consideration that, probably, helped this word survive thousands of years and still stay in use today is its root. It is the same as in לְהַרְכִּיב to assemble
and
מֻרְכָּב complicated
So you already can feel that the associations involved are those of something not simple, something that consists of many parts. Perfect word to choose when something needs to be fixed! ;) And this is exactly the word you would choose if you wanted to sound more serious, more respectful of you vehicle and, maybe, more adapt in how exactly it functions.

So, finally, some examples:

אֵיזֶה אוֹטוֹ זֶה? גָּדוֹל אוֹ קָטָן? גָּדוֹל, נָכוֹן! וּמָה הַצֶּבַע שֶׁל הָאוֹטוֹ? יָרֹק, נָכוֹן! כָּל הַכָּבוֹד, אֵיזֶה יֶלֶד חָכָם!
(What car is that? A big one or a small one? A big one, right! And what color is the car? Green, right! What a smart boy, good for you!)

מֵחַלּוֹן הָאוֹטוֹבּוּס רָאִיתִי יָם שֶׁל מְכוֹנִיּוֹת לְפָנֵינוּ - פְּקָק בִּכְבִישׁ מִס' 1 זֶה דָּבָר רָגִיל בַּשָּׁעוֹת הָאֵלֶּה.
(From the bus window I saw a sea of cars in front of us - a traffic jam on highway #1 is a normal thing at these hours.)

מַשֶּׁהוּ לֹא בְּסֵדֶר עִם הָרֶכֶב, אֲנִי צָרִיךְ לִמְסֹר אוֹתוֹ לַמּוּסָךְ בַּשָּׁבוּעַ הַבָּא.
(Something is not all right with the car, I have to take it to the garage next week.)