Argthtjtr

I often see people on the Internet using a comma before and in many cases (not adversative cases). Is it ok? In my language it is stricly prohibited to use a comma before an and except for adversative cases or when an apposition is in the front of that and.

Examples (which I consider not ok):

He is a great player, and he prefers to play Counter-Strike.

John joined the Army, and George joined the Marines.

What I consider exceptions:

Yes, and what is the problem?

I called John, my brother, and now I’m speaking with him.

From the Oxford Guide to Style 2nd ed section 5.3:

Use the comma to join main clauses that are semantically related,
grammatically similar, and linked by one of the coordinating conjunctions
and, but, nor, or, and yet. Such clauses are joined by a comma if they
are too long, and too distinct in meaning, to do without any punctuation
at all, but not separate enough to warrant a semi-colon:

Truth ennobles man, and learning adorns him.

Cars will turn here, but coaches will go straight on.

I will not try now, yet it is possible I may try again in future.

It may be omitted when the clauses are short and closely linked:

Do as I tell you and you'll never regret it.

Dan left but Jill remained.

I will not try now yet I may in future.

All four of your examples are compound sentences, which permit linking commas:

A compound sentence is composed of at least two independent clauses. . . . The clauses are joined by a coordinating conjunction (with or without a comma), a correlative conjunction (with or without a comma), a semicolon that functions as a conjunction, a colon instead of a semicolon between two sentences when the second sentence explains or illustrates the first sentence and no coordinating conjunction is being used to connect the sentences, or a conjunctive adverb preceded by a semicolon.

As @aurora notes, we commonly use commas with coordinating conjunctions except when linking very short and closely-related clauses, and even then it's never incorrect to use the comma. Here's how your sentences parse, with some additional notes.

Clause 1: He is a great player.
Clause 2: He prefers to play Counter-Strike.
Compound: He is a great player, and he prefers to play Counter-Strike.

Because both clauses have the same subject, you could could naturally replace and he with a relative pronoun: “He is a great player who prefers to play Counter-Strike,” making it a complex sentence instead of a compound.

Clause 1: John joined the Army.
Clause 2: George joined the Marines.
Compound: John joined the Army, and George joined the Marines.

This is a straightforward example of a compound sentence. Because both clauses are short and similar, you could omit the comma if you prefer.

Clause 1: Yes.
Clause 2: What is the problem?
Compound: Yes, and what is the problem?

Here, Yes is a special kind of clause called a pro-sentence.

Clause 1: I called John, my brother.
Clause 2: Now I'm speaking with him.
Compound: I called John, my brother, and now I’m speaking with him.

In this case, the phrase my brother is in apposition, so it must be set off with commas regardless of the structure of the rest of the sentence.

English doesn't have any formal or official rules, so nothing can be prohibited at all, certainly not strictly prohibited. It does, of course, have generally accepted modes of usage, which may be considered as informal rules - but the important thing is that the 'rules' are determined by usage, and not vice versa.

The issue in determining comma usage is whether the presence of a comma helps to break up a long sentence, whether it helps the reader in understanding the sentence, whether it helps to indicate a natural pause in the sentence.

The only one of your sentences that I would change is the first:

He is a great player and prefers to play Counter-Strike™.

(I have been informed that "Counter-Strike" is a Trade Mark and should therefore be capitalised. I have no information as to whether it is registered in any country.)

Note that I also omitted "he". Having said that, there is nothing wrong with your original version, just that the second version may sound more natural to a fluent speaker.

You should also look at the other questions links from the comments. Although several of them may relate to lists, they may still have useful explanations about comma usage.

As others have written, comma usage is not required in some scenarios; commas should be used in compound sentences, where there are two independent clauses being combined, but they are optional in coordinating conjunctions (Oxford comma). A quick Google search reveals this information.

In other scenarios, we may or may not opt to use the Oxford comma. The most important thing is that the writing is consistent; writing that includes coordinating conjunctions both using and not using the extra comma can confuse the reader. It is important to research the format you are writing in, as different writing styles may or may not mandate the usage of the Oxford comma. For example, the Chicago Manual Style requires the Oxford comma while the Canadian Press forbids it.

As you guessed correctly, there are some exceptions to where commas must be placed. For a more detailed summary, view the Wikipedia article.

A programming manual describing the logical operator 'AND' would have a comma before that word and be grammatically correct. I know it's a cheeky answer, but it's an answer :).

Digital Logic operators include OR, AND, XOR, and others.