Hard to say with which verbs 把 is used most often. However here are some examples to your other question (when to use it). Compare the following example:
The meaning is "I ate your egg soup" in both cases. However, in the first case the emphasize is on eating the soup while on the second one it's about the result of the soup being eaten already. Look at this 2 dialogues:
A: 你吃了吗？ (Have you eaten?)
B: 吃了。我吃了你的鸡蛋汤。(Yes. I ate your soup.)
Topic of this conversation is "eating", what you've eaten is not important. Maybe there is even some soup left.
A: 我的鸡蛋汤呢？ (Where is my soup?)
B: 对不起。我把你的鸡蛋汤吃光了。(Sorry. I ate it up.)
In this conversation A wants to know where his soup is. Topic obviously is the soup itself, if you or someone else have eaten it doesn't matter here.
我把头发剪短了。(I cut my hair short.)
Emphasizes: hair is short now. Who, when, how it was cut doesn't matter.
我把邮票贴上了。(I put the stamp on.)
Emphasizes: stamp is put on. Doesn't matter when you put it on, how you did it, ...
Generally I think one could say: 把 is used when you want to emphasize on the result. Same goes for
请把门关上。 (Please shut the door.)
Literally this is: Please make door shut, which is exactly the meaning of it. Just make it happen that the result (door closed) is accomplished. If you actually go up and pull the handle or swing your wand to close it doesn't matter.