Standard English grammar is to capitalise proper names, but not nouns. the word "the" may be used to preface a noun, but not a proper name (except to distinguish which "John Smith" one is talking about - "The particular John Smith who ate Freddie Starr.") Note that a name may include "The" (capital T) as in "The Dog and Duck."
As such, the grammatically correct usage would be "the internet". The accepted usage of "the Internet" is irregular, so one may refer to "the internet" whenever one wishes to set a trap for a pedant.
email or emails? Generally, the collective "mail" should be used.
However, unlike snail mail the word is sometimes used to refer to an individual message: one might say "I opened an email the other day," but never ""I opened a (snail) mail the other day." In this context, one should use "emails" to refer to several messages, as: "I opened three emails the other day."
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