May I ask you a question about the difference between different "left" and "rest"?

For example, when I go went to a the restaurant and I'm already full. 
That time, could I have said can I say to a staff member stuff of the restaurant that ...? 
1. “Can Could I get the rest of this to take home?” 

  (You would say the 1st one)
  (You can also use left in this situation with some rewording, but it is more common to use rest)

Another one,

I came back to bring a dog who was abandoned by to someone home, but I couldn't keep do it as a pet because my mother wouldn’t didn't permit it. I can't help it. 
Then I brought the put a dog to the park. 
That time, could I have said can I say…

1. It I seemed that the rest of the dogs felt sad. 
2. It I seemed that the dog who was left in the box felt sad . 

(Either can work in the situation)

Another one,

when I had to have prepared a dinner, I checked the a fridge but there wasn’t much is seldom food inside. 

 (Seldom in this case is a bit awkward)
That time, could I have said can I say...? 

2. I prepared a dinner using the leftover food in the fridge.

  (or “I prepared a dinner using the food left in the fridge.”)

 (Once again, you can also use "rest" in this situation with some rewording)

 Overall, I think that the difference between “left” and “rest” is simply the syntax of "the sentence. "Rest" is usually "the part that is left or remains; the remainder". "Left" is usually "to allow to remain in the same place, condition, etc". Sometimes the sentence is easier to construct using “left” {Example 3} and other times “rest” is easier to use {Example 1}. 

I hope this helps, sorry if it didn't.