Learn English – the meaning of updoots

NOAD lists three definitions for shoulder (I've omitted Definition #1, as that is the classic anatomical shoulder):

2 a part of something resembling a shoulder in shape, position, or function : the shoulder of a pulley.
• a point at which a steep slope descends from a plateau or highland area : the shoulder of the hill sloped down.
3 a paved strip alongside a road for stopping on in an emergency.

With those definitions in mind (particularly the subdefinition under Def. 2), it seems like shoulder tells a bit more about the hill than side might; specifically, it implies the top of the hill is relatively flat, with a relatively steep slope downward from the top. Depending on their shapes, some hills might not have shoulders.

In regards to the sentence, I think the shoulder of the hill is a more precise area than the side of the hill. Looking at the image below, I there are several hobbit holes in the side of the hill, but only the ones boxed in orange are ones that I would place on the shoulder of the hill, that is, in the area where the slope descends downward from the highland area:

Unless Mr. Baggins lived in one of the lower holes, I wouldn't really call this a "mistranslation" – just a word that is perhaps more detailed than it needs to be.

The meaning of "spring in one's step", as Manny used it, is this:
Wiktionary

spring in one's step (idiomatic)
enthusiasm, energy or a positive outlook or cheerful attitude.
After her promotion, she carried out her new position with a spring in her step and a contagious smile.

The meaning of "spring" is "jump" or "leap". Someone who has a spring in his step is the opposite of someone who is depressed.

However, Jay has interpreted "spring in my step" to mean that Manny's peers have told him he has a spring in his step because they think he is effeminate, or even homosexual. Manny didn't mean it that way at all, and "spring in my step" is his own phrase, used to mean that he is light-hearted because of his infatuation with a girl. The joke is Jay's misinterpretation, contrasted with Manny's unconcern.

Compare "light in the loafers":
Grammarphobia

Cassell’s Dictionary of Slang (2d ed.) has only a brief entry, describing the expression as ‘50s American slang and adding that “the image is the stereotyped effeminate male, tripping along.”

The Random House Historical Dictionary of American Slang, which defines it as effeminate or homosexual, lists a series of references for the expression dating from 1967 to 1996.