Terminology – What Is It Called When Two or More Nouns Come in a Row?

Most of the time, the names aren't particularly inventive.

"Шапка-невидимка" — a hat which, being put on, makes a man invisible;

This would probably be an invisibility hat or hat of invisibility, compare to a cloak of invisibility (which is quite a common item).

"Сапоги-скороходы" — self-running boots;

Compare with seven-league boots, which make each step cover 7 leagues of distance (38.8km). Also compare with the winged sandals of Hermes in Greek mythology.

"Скатерть-самобранка" — a table-cloth which produces all sorts of meal and drinks according to its user's preference;

I don't know of anything common that's similar to this, but it did remind me of the Land of Goodies from Enid Blyton's The Faraway Tree books – the Land of Goodies is a place where everything is made of delicious food.

"Ковер-самолет" — a self-flying carpet;

Just a "flying carpet" or "magic carpet" in English. I imagine most people would think of Aladdin when you mention flying carpets.

"Живая и Мертвая вода" — the two kinds of water, one called "alive" and the other — "dead". The former heals the hero's wounds, the latter revitalizes him;

I haven't heard of anything like this. Something more general would be potions, magic concoctions which may have any number of effects. The term "elixir" is also for liquids which heal and give strength.

"Молодильное яблоко" — an apple which, having been eaten, turns the old into the young.

Not a food, but compare with the Fountain of Youth, which is a fountain which makes young anyone drinking or bathing in its waters.

As for common English folklore, a very common story is Jack and the Beanstalk, where a young boy (Jack) barters for magic beans. During the night following the beans being planted, a giant beanstalk grows. At the top of the beanstalk, Jack finds a large castle inhabited by a giant. Jack steals from the giant, and the giant chases him down the beanstalk. Jack then fells the beanstalk, killing the giant and living happily ever after on his ill-gotten riches.

The phrase "happily ever after" is also a common element in many English-language folk stories.

An element in Scottish folklore is the kelpie, a water spirit that lives in lochs that is said to have the strength of many horses. These are particularly noteworthy due to the massive statue called The Kelpies that was built in Falkirk.

Based on the above dictionary definitions, it seems it is valid to say "English is my target language." However, I do not believe that this expression is in common use and it does not sound entirely "correct" to my ears.

If I heard someone say "English is my target language" with no context, I would not immediately jump to the conclusion that he was learning English as a second language. I'd have to ask for clarification.

In my mind, the word target does not go well with language. When target is used as a modifier, it implies that an action is directed toward or at that object.

For example, the target audience is the audience that you are directing the action of talking (most likely but it could be other actions like selling) to. The target voter block is the set of people you are trying to appeal to. The target school is the school you are directing the action of "trying to get accepted" at. (I was actually having trouble coming up with nouns that "target" could modify).

When target is used to modify language, it almost sounds like you are trying to direct an action at "language". But "learning" which is the action that would be understood to be directed doesn't really have that feel in English. English considers "learning" to be more of a passive action.

Just a disclaimer, and anyone please comment if you know better, but this is entirely speculation made up on the spot by me.

That being said, I know that "English is my target language" is not a good way to say that you are learning English.

With regard to your other question, I do not believe "target language" is a common expression used in English at all (Regardless of formality).