First - Snuggling vs. Cuddling
Snuggling and cuddling can mean the same thing which is something like "hugging" for comfort or intimacy. But "cuddle" - while it can be mutual - often implies one person cuddling another. That is, there's can be a sense of a "cuddler" giving comfort to a "cuddlee". The following is mostly described as "a father cuddling his baby":
And here, our feline friends "Odin and Ruby are snuggling". Also, Ruby (the cat on the right) is snuggling his nose into Odin's white fur. And just like Ruby, a single person can feel good just "snuggling under the covers". But one could also say, "these cuddling cats are cute".
Now the Answer
Let's assume your scene is a mother talking to her child. The mother is sitting next to her young daughter who is near a doll. The mother is offering some consoling words to the daughter, who is afraid of the dark.
A) "If you're scared, snuggle with that doll, baby." is the most natural of the three you presented. This would be especially true if the doll was rather large and already next to the child. It implies a mutual snuggling, just like Odin and Ruby, above. That would be exactly the correct thing to say to a young child, since the idea of the doll being animate and being able to provide comfort back is what provides comfort to the fearful child.
B) "If you're scared, snuggle that doll, baby." could sound a bit too curt - like an order. However, there's a type of mother-character one might see in movies (or real life I suppose).. the no-fluff, down to earth, common-sense, yet-loving mother that might deliver the message like this.
C) "snuggle up to" focuses more on the movement, and may suggests a more playful action. (Also, between adults, snuggling suggests more intimacy than cuddling.) This would be the least natural, in this case, in my opinion.
The difference between using "snuggle vs. cuddle" with a child would mostly depend on what the child already thinks of as being comforting. It's not the specific word, but rather, which word the child would feel the most comforted by. One could suggest that a child cuddle her doll (like the father above) or cuddle with her doll (like the cats) for comfort, if that's what she is used to doing and calling it.
FYI: This is also reminiscent of the Harlow Monkey Studies.
Google Ngram: snuggle *,cuddle *,snuggle that,snuggle this,snuggle me,cuddle that,cuddle this
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]
- 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.
- 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."
Best Answer
Prepositions can be tricky because they often have multiple meanings depending on context.
"By" can mean "near", as in, "The house is by the water." It can also mean "using", as in, "We travelled by car." That works here. "We shared by email" means "we shared using email".
"Through" usually means "passing over the boundaries of", as in, "We drove through Spain". It carries the implication of going in and coming back out -- you don't use "through" when you stop inside the place or thing. You might use "into".
We sometimes use "through" to describe a method of communication, I guess on the thinking that the message travels "through" this communication system. Like, "I sent a letter through the mail", as in, the message travelled through the postal system. Or, "We kept in contact through the telephone." So in that sense you can say, "We shared through email", i.e. the sharing went through the email system.
Note this usage is pretty much limited to communication. You wouldn't say, "We travelled through train", you'd say "We travelled by train." ("We travelled through a train" would mean that you walked up and down the length of the train, or that you walked in one door of a train and then out the opposite door, not that the train carried you somewhere.)