Learn English – When to use hurt vs hurts

"Shout" and "yell" are more or less synonymous. Other than style or poetry (and a few idiomatic expressions), I can't think of any reason why you couldn't use one in place of the other.

A "scream" includes a shrill note that indicates extreme emotion. This is the infamous "Wilhelm Scream'.

These are also screams: American girls screaming at the Beatles

A "cry" is a loud exclamation, often sounding like someone (or something) in distress. There is also the additional nuance that a "cry" is meant to be heard over some distance.

As you mentioned, you can "cry" for help, but so can animals: Bear cub crying for its mom.. We can describe the sound wildcats make as "cries" (or "screams") even though these are not from distress, because it sounds like something that needs help.

(Edit) We would most likely say that a heavy metal vocalist screams into the microphone. Apparently this (along with growling) is the actual term for this style. As a scream, by definition, includes higher-pitched elements, a rapper who is "singing" loudly into the mike could be said to be either shouting or yelling.

Be aware that, in current slang, to "shout out" means to "favorably acknowledge" as in "a public expression of thanks".

I want to give a shout out to all of my fans, without whom I wouldn't be here on this stage.

I appreciate your raising this intriguing question that has some important implications for how or why a particular expression or construction is considered grammatical. Merriam-Webster dictionary clearly gives the example "requested her to write a paper" under the transitive verb definition of 'request' at https://www.merriam-webster.com/dictionary/request (scroll down past the extensive 'noun' definition to read) so the form 'request someone to do something' is definitely grammatical from a technical perspective.

However, it is no longer a popular usage: according to google ngrams, the popularity of both "request you to" and "request that you" peaked around the year 1830 and declined steadily thereafter. As you rightly noted yourself, "request you to" enjoyed a brief resurgence peaking around 1970 but both expressions became progressively unpopular (rarely used) in the last quarter of the 20th century, indicating that such extremely formal language has lost favor in English all aroung the world, although "request that you" is marginally ahead at the turn of the millennium:

That is not to say that such expressions are equally outdated everywhere. Based on the answers to a number of related questions on ELU and ELL, typical patterns and frequency of use seem to differ between dialects such that "request someone to do something" is more common in Indian English and "request that someone do something" is more typically used in the USA.

The extensive explanations of senior member @tchrist both in comments here and in an updated answer on ELU present a convincing argument (supported by ngrams) that the construction "request someone to do something" has become obsolete in modern American English. As regards British English,

If you swap American for British you can see what's happened: it has not receded quite as much in British writers as it has in American ones. This might be because British don't use mandative subjunctive as much as we do [...] - tchrist see full comment here

If "grammatical" is defined as "the way native speakers are using the language now" then the extreme rarity of such usage in the case of 'request someone to do something' might well lead it to be considered ungrammatical by speakers of American English. Whereas the same form is so commonly and typically used in Indian English that as an Indian I find it extremely natural to the formal register, and unquestionably grammatical.

As a transitive verb, "request" can take an object in multiple ways.

(1) Ask for something:

She requested a room with a view.

The librarian requested perfect silence.

Here "request" literally means "ask for" in a formal/polite sense. Whatever is being asked for is the object of "request."

(2) Ask for something from someone:

May I request your attention.

He requested the students' attention.

Here somebody is being asked politely for something but the object of 'request' is still "attention."

(3) Ask someone to give you something:

"May I request you to give me your passport", the customs officer said. _{(Same basic meaning as "may I request your passport" but this is not really a request but a command, and 'may I' is used simply for politeness here. Based on user feedback in comments, most Americans would prefer "may I request that you show me your passport" as the most politely formal construction. )}

He requested the students to give him their attention.

Here the object of 'request' is "students"; the students are being asked to give him something, and the object of 'give' is "attention." However, he is essentially asking for their attention and this is just another way of writing "He requested the students' attention."

(4) Ask someone to do something:

He requested the guests to leave their cars at the gate.

Here the object is 'the guests' and he is requesting them to do something. This can be rewritten in passive voice as

The guests were requested to leave their cars at the gate.

Whereas the alternative form "request that" sees 'request' used with the conjunction 'that' which connects it to the clause that follows:

He requested that the students give him their attention.

May I request that you bring your own dinner.

He requested that the guests leave their cars at the gate.

The host requested that cars be left at the gate.

Both these forms of 'request' are nicely described at Collins Dictionary online:

Request [verb]

1. If you request something, you ask for it politely or formally.

Mr Dennis said he had requested access to a telephone.

She had requested that the door to her room be left open.

2. If you request someone to do something, you politely or formally ask them to do it.

Students are requested to park at the rear of the Department.

They requested him to leave.

Source: https://www.collinsdictionary.com/dictionary/english/request

Note: both "may I" and "request" as used in these examples are politely formal ways of asking for something, or asking someone to do something. In this type of usage a question mark is not necessary even though the sentence begins with "may I."