There are many confusing rules in grammar use. When you use an apostrophe (who's) in this instance it can only be the contracted form of "who is." If you do not wish to say 'who is' then your choice would be "whose."
To refer to the sentences you used:
1) "And who is getting the pizza?"
This sentence would use "who's" because it is the shortened or contracted form of 'who is.' This is obviously a question so it makes "who" an interrogative pronoun because it is asking the question "which person?" You should also not start a sentence with the word "and" as it is a conjunction so should be used between sentences.
"Who's getting the pizza?"
2) "Who's turn is it to help those whose grammar is 'iffy?'
By now, you can probably see the first error in your second sentence as we do not want to say "who is" so must use "whose"at the beginning of this sentence. The second whose in the sentence is correct. The only difference is what part of speech each word belongs to .
The first (and corrected) whose in this sentence is an interrogative pronoun. The second 'whose' is an interrogative adjective.
As you are probably not concerned with what we call them and whether they are interrogative pronouns or adjectives and so on, the rule of thumb should be to check if you want to use "who's" to say the same as "who is" in which case use "who's" and if you don't want to use it to mean 'who is' then the only choice is "whose" no concern as to whether it is adjective or pronoun.
Who's going to give me a thumbs up for this answer?
Whose party shall we attend?
I don't know whose marks are the best.
Who's worried about those whose grammar is not quite perfect?
It's a question of who's who.
When is a "whose" not a "whose?"... When it's a "who's....."