"Answer to me"
This isn't obvious from the words but a boss would say to someone below them "you answer to me" as in "I have the highest authority, not anybody else". I would say this comes from military speech; as in when a higher ranking member of the army says "Private Smith" and the lower ranking member goes "Sir, yes sir", so you answer to him, because he is your superior.
"Answer me" means what it says. If I asked you a question and you stayed silent I would say "answer me", to try and prompt you to answer.
"Propose to me"
The most common is a marriage proposal- "He proposed to me last night and I said yes!".
or also as another kind of proposal- "Could you propose to me a better title?"
"propose" can mean to present or state something, so you could use it like; Person1-"They want me to put someone forward for the job" Person 2-"You should propose me for it".
Not 100% sure on that last one, just because propose has a few meanings and it's tricky to know what one they mean without context. :)
(I'm not a teacher btw, just like to help people out.)
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