Normal and inverted exclamation and question mark pairs

Retina, 39 37 34 bytes

\w[^.]*?([!?])
$1$0
T`?!`¿¡`\S\b

Try it online.

Explanation

\w[^.]*?([!?])
$1$0

This matches a sentence ending in an exclamation or question mark, and prepends that punctuation character to the sentence. Now we know that all the ! or ? which are immediately followed by a non-space character must be those we inserted, because the original ones should be separated from the next character by a space.

T`!?`¡¿`\S\b

This transliteration stage turns all ! and ? into ¡ and ¿, respectively, provided they're found in a match of \S\b, which applies only to the ones we just inserted. Replacing both in two separate substitutions in the same byte count, but I prefer the semantics of a transliteration stage here.

Javascript (ES6), 86 79 66 63 bytes

i=>i.replace(/\w[^.!?]*[!?]/g,k=>(k.slice(-1)>'>'?'¿':'¡')+k)

Ungolfed:

func = inp => inp.replace(/\w[^.!?]*[!?]/g, sentence => (sentence.slice(-1) > '>' ? '¿' : '¡') + sentence)

Usage:

console.log(func(`Hello there!      What is your name?
My name is Ron. What's your name?
My name is Alex.  Nice to meet you!
Nice to meet you to!  How was your break?
It was great, I spent all my time code golfing!
What's that?
            Wow, you're such a n00b! Here, let me show you.`))

Will implement bonus solution soon.

Thanks to:
@user81655, 86 => 79 bytes

Mathematica, 101 92 91 bytes

StringReplace[#,RegularExpression@"[A-Z][^.]*?([?!])":><|"?"->"¿","!"->"¡"|>@"$1"<>"$0"]&