I think this dream means that there is something negative in your life that you want to get rid of, but you are afraid of what else you might have to sacrifice or of having the responsibility for taking any action.
In general, any type of
monster in a dream can represent something negative in your life, often a negative quality that you have. The fact that the monster is a giant can mean that you feel that this negative quality has gotten too big to handle.
It's interesting that in the dream you get help from your father and your boss, both authority figures who can have a parental role. At the same time, you act dead and later collapse. This can be a sign that you don't want to take responsibility, that you want someone else to take care of you or to take care of your problem for you.
There are a lot of dogs in your dream. A
dog can stand for your personal boundaries or represent loyalty. When your partner says that you have to sacrifice the dog, you may feel that you are going to have to give up an important part of yourself or betray yourself or someone else in order to achieve your goal of doing away with this negative influence in your life.
However, it seems that this is a mistake because your partner crashes the car anyway, so getting rid of the dog doesn't help.
And later, there are more dogs in your dream, first your family's 2 dogs, and then the picture of Lassie.
So you don't really have to give up that much.
Falling down
stairs represents the doubts that you are having about all of this.
In the end, you get to where you need to go on your own, by jumping over fences, which is a sign that eventually you can get to where you want to be on your own.