"lait" in uncountable so either you drink milk or you don't "boire du lait" - "ne pas boire de lait" (cu changed to de in a negative sentence).
Sandwich is countable, you could eat one or several if you wanted to. In that case, I think that both singular or plural are acceptable, depending whether it is a punctual fact or a generality (more probably with "jamais").
No, the only change would be "sandwich" is singular. The issue here is to understand that "pas de" may introduce either a plural or a singular. The question you may ask yourself is "if the sentence was in positive, rather than negative, would the object be plural or singular?".
Sometimes, the answer is clear: - je n'ai pas de voiture (otherwise I would have one) - je n'ai pas de fleurs dans mon jardin (otherwise, I would have more than one)
To add to what Sitesurf said, it's basically because of the "never." While in English you might say, "I never want to eat a sandwich," you're still really talking about more then one sandwich (any sandwich that comes along ever) because of the use if the indefinite article along with never. It's a tricky subtlety. French requires that you express this with a plural rather than literally translating it.
When the sentence is not negative, this is what you get:
- je veux manger un sandwich (singular) -> je veux manger des sandwichs (plural)
In a negative sentence, "des" (plural of "un") is changed to "de".
- je ne veux pas manger de sandwich (singular = right now) -> je ne veux pas manger de sandwichs (plural = in general).
Because it's English, not Latin. The rule against splitting infinitives was invented out of whole cloth by Robert Lowth and other grammarians in the 18th century because they wanted to revise English to be more like Latin, not because they were recording the rules of actual English. Splitting infinitives in Latin is literally impossible; they're single words. It makes no sense in English morphology to prohibit it (English uses auxiliary prepositions), except to fulfill that revisionist goal. In any case, it's not how people actually use English.
It has nothing to do with Latin. It has to do with "exceptions" requiring by hand approval from site surf or the (newly added?) other admin (sorry. Forgetting name right now). The people who "base code" the English are native French speakers, with a distaste for split infinitive via their schooling. They have no ability to fix things from the forum. You have to report the issue inside the question (my answer should have been accepted).
Sorry if I wasn't clear, but I was only referring to the rule that one shouldn't split infinitives in English, not the approval process for answers in Duolingo. That a native French speaker would find it strange is certainly understandable, but that only seems to underscore the fact that Romance language rules applied to English aren't always indicative of what's correct or what should be acceptable. If the reason that a split infinitive form of the sentence hasn't been added has more to do with a distaste for split infinitives than it does with (what I'm sure is) the slow, iterative process of covering all the bases, then I'm saying that reason should be put aside.