Learn English – Is this expression formal or informal

Well, here is how you can tell those sentences in a formal way. Note that there could be many ways but I'm writing what I find closest.

You don't say - used when you agree with someone, or surprised, or utter this in sarcastic way.

"His behavior is very strange." - Yeah! I agree (agreeing).
"Jack won the championship." - Ah, did he? Don't tell me (surprising).
"You know, there's no sun in the night." - Wow, I never knew that! (sarcastic).

Challenge accepted - the speaker is ready to accept the tough task that you think is not possible.

"It's impossible to finish this task by 1800 hr." - Do you think so? Well, I'll do [finish] it.

Bitch Please - A response used when someone says something stupid or when somebody tries stupid on you

"...so you understand the scheme? If you give me 100 bucks, I'll make it double in a week." - Ah, come on! Stop this.

RIP [somebody] - Rest In Peace. This is said when a person has died and you pray that their soul rest in peace - This is abbreviation rather than informal way of expressing grief. You may use the whole phrase and it's okay.

However...

RIP [something] - a tongue in cheek comment when something is no more in the market, internet or the like - Here, you don't need to be formal.

Check this -

In spite of is not an idiom but a prepositional phrase. To alter (something) beyond recognition is just one of usage of the verb alter. They shold not be classified as idioms because of the following reasons:

Idiom is defined in Merriam-Webster as:

1. an expression that cannot be understood from the meanings of its separate words but that has a separate meaning of its own

2. a form of a language that is spoken in a particular area and that uses some of its own words, grammar, and pronunciations

3. a style or form of expression that is characteristic of a particular person, type of art, etc.

Idiom is a very-well known word, but some people don't understand what it exactly means. According to the definitions above, those expressions/forms of language/styles and forms of expression should not be written in a formal statement.

However, when it comes to mean in a broader way:

a common word or phrase with a culturally understood meaning that differs from what its composite words' denotations would suggest.

It can be used in a formal statement. They are broadly used by many politicians and businessmen.

Here is a list of English idioms in Wikipedia. You will find many of them are very familiar to you and useful. And you will clearly see how the listed idioms are different from in spite of and to alter (something) beyond recognition.