Learn English – Is “plugging out” electronic devices an American expression

Intonation is a general term for the pitch of your voice rising and falling as you speak:

• It includes things like the pitch of your voice rising at the end of the sentence to signal a question, or lowered pitch to indicate a parenthetical phrase.
• It excludes phonemic pitch contrasts used to distinguish words as in tonal languages (Thai, Mandarin) or pitch accent languages (Japanese)—although you might find some definitions that include this sort of thing.

Accent has a few different meanings. The relevant meaning here is "features of pronunciation that signal regional or social identity" (Cambridge Encyclopedia of Language, p.420). This could include differences in intonation between different groups of speakers, although I wouldn't go so far as to say it includes intonation generally.

Your examples of eye dialect appear to describe differences in vowels between different groups of speakers. This sort of difference falls under accent.

Dialect, by the way, is different from accent. It's defined as "a language variety in which the use of grammar and vocabulary identifies the regional or social background of the user" (Cambridge Encyclopedia of Language, p.425). So a word like prepone existing in Indian English but not American English is a matter of dialect rather than accent.

I had hardly come in when it started to pour with rain.

No sooner had I come in than it started to pour with rain.

Basically right, but there are a couple of non-idiomatic elements in those examples.

The first sentence is a bit off. As @oerkelens points out, "with rain" is not usually heard in such statements. And "come in" with "hardly" is not the right expression for getting into the house in the nick of time, though it does work with "No sooner had I come in..."

Here in Philadelphia, Pennsylvania (where there's an Irish-English influence) we might say

I'd barely made it in the door when it started to pour down rain.

or

I'd hardly gotten through the door when it started to pour.

or

No sooner had I made it through the door than it started to rain cats and dogs.