Vamos means "we are going to" - it's in the affirmative.
"Let's sell the shoes" is a suggestion. It can be used colloquially to be an affirmation, exactly like the idiom, "Let's do this!" Though, if you are speaking with strangers or with even slight formality it wouldn't be a definite order like, "we are going to"
Also, depending on context, "we are going to" can be pointing to a less strict timeline future scenario. Whereas, "let's" would still be a suggestion.
"Let's sell shoes." "Good idea! We are going to sell shoes."
"We will sell shoes" = "Venderemos zapatos" "Los zapatos" refers to specific shoes, the ones that yo have already talked about. "Zapatos" without the article means shoes in general; you are going to into the business of selling shoes. The articles are not superfluous or redundant.
My understanding is: If it were "...sell red shoes", it would be "...los zapatos rojos" Spanish uses the article when the direct object (shoes) is modified).
However, in this sentence, there is no "los" because a definite article is not used with an indefinite amount, an unknown quantity. If you can put the words "some" or "any" before it, it is an indefinite (unknown) quantity.
The second part of your explanation is correct, but the first part isn't. Modifications don't influence the use of the definite article. (Only of the indefinite one in, for instance, job descriptions.)
"We are going to sell red shoes" would still be "Vamos a vender zapatos rojos." If you say "los zapatos rojos" it would again be "the red shoes".
'We ARE GOING TO sell the shoes' seems like a more accurate translation from what I understand here. If one replaces 'are going to' with 'will', won't that change the Spanish sentence? I sometimes think in Spanish without realizing it and type spanish when I am supposed to type english^^ I guess it's a good sign that I am learning alot:)
Les is not an article. The definite articles (that translate to "the" in English) are el (masc sing), la (fem sing), los (masc plur), and las (fem plur).
Les, on the other hand, is an indirect object pronoun, and translates as "to/for them", mostly: "Les di buena comida" - "I gave good food to them."
In Spanish, 'b' and 'v' make the exact same sounds, but two different ones. At the beginning of a sentence, after a pause, or after 'm' and 'n' it sounds like 'b', everywhere else like a softer 'v'. So the sentence
Bebemos vino en invierno. (We drink wine in winter)
is going to sound more like
"Bevemos vino en inbierno."
"We are selling the shoes" is present progressive tense (which Spanish has as well), but the construction "ir a [verb]" denotes the phrasal future tense, a pretty direct translation of the English "to be going to [verb]".
"Going to" future and "will" future are often interchanged in this course (which I'm not a big fan of), so this sentence allows both the translations "We are going to sell the shoes" and "We will sell the shoes".