Learn English – for a long time / for a while

It's not common to use "after a long interval" to mean "after a long time." Your sentence "I had sushi for the first time in a long time" sounds perfectly natural.

Long time no see probably derives from pidgin English spoken by Native Americans or Chinese immigrants, although no one is completely sure.

It matches the Mandarin Chinese phrase 好久不見 (hǎo jiǔ bú jiàn) word-for-word,* which is grammatical in Mandarin.

When you say long time no see, you are pretending to speak broken English, fitting English words to a foreign or pidgin grammar. The phrase is used so much, many people use it without knowing that, but to native ears, it certainly sounds like pidgin English because of the ungrammaticality you noticed. A few other pidgin phrases have gained some currency in English, such as: look-see, from Chinese pidgin English, as in "I'll have a look-see"; savvy to mean "understand", as in "You savvy?", probably from West African pidgin English; and da kine, Hawaiian pidgin English for "the kind", but meaning pretty much anything, or sometimes "the original kind" or "the best kind". To retain their distinctive character, these phrases don't adapt to English grammar; they retain their own grammar even when included in English sentences. Sometimes people modify long time no see, retaining the pidgin grammar; for example, long time me no see you girl.

^{*Ignoring the usual subtle differences between corresponding words in different languages.}