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
Both ways of writing the sentence are technically valid, but things change a bit between the two in their meaning.
What reason would your father have to lie?
Think of this as, "What reason would your father have, to lie?" Here, "would ... have" acts as the verb, with "what reason" and "to lie" both technically acting as the objects. ("to lie" may be a different part of the sentence, but I cannot remember the exact term for it.) Note that in this case, it is implied that the father lied willingly.
For what reason would your father have to lie?
Here, the complete phrase "would ... have to lie" is the verb, with "what reason" being the object. In this case, it is implied that the father "had to lie" or was forced to lie.
If I remember the movie correctly, the first case implies the correct meaning, so the original movie line is correct.
Best Answer
This phrase doesn't really mean "I don't want any lecturing," because that sounds kind of negative. "You had me at hello," is always positive - it means that I agreed with you (or even fell in love with you!) at the very first instant you spoke, and nothing you said afterwards can make me feel any more positive towards you or whatever idea you're promoting. So while this phrase doesn't necessarily mean "charm" in the romantic sense, it's definitely more positive than just "to persuade," which is why @user45266 used "you won me over" as a synonym.
The reason why at is part of the catchphrase is because what you are emphasizing is the position of the words that won you over (i.e., the very first one or ones), so you use at to mean "expressing a particular location or time." Using with is grammatical, but does subtly change the emphasis to stressing which word it was rather than stressing the fact that it was the opening words.
If you want some equivalent expression that's less positive, you could say, "I got it already," which says that you already understand something and can be used in an exasperated way to mean, "I don't want any more lecturing."