I think the answer to this question is fairly obvious.
"Where there is no trust, there is no love" refers to the fact that a relationship is not a loving one if the people in that relationship cannot or at least do not trust one another.
love is irrational and stupid ;)
It's not an idiom because you can figure out the meaning by thinking. A PhD is a college degree meaning you have studied a subject intensively and know a lot about it. If you are a doctor of love, you know a lot about love.
I believe the idiom you're looking for is "You can't trust a person further than you can throw them". The meaning is meant that you cannot trust someone because you can't throw them very far.
To say that there is no love lost between two people means that they dislike each other.
"To be" is not an idiom - it's a verb.
Pest is not an idiom. It's a word.
The idiom "apple shiner" means the teacher's pet.
The meaning of the idiom in the pink of health means being in good health.
This means that you can trust that person.
The idiom means impress someone is egg on
It's not an idiom - to cope means to deal with, or to handle
"Old hand" is an idiom meaning having lots of experience.