1. I will take him home with me. (him=a dog)
2. I have no money with me.
3. She lives with her parents.
4. I will take him home.
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Q1: What is the difference between #1 and #4?
Q2: what is the meaning of 'with' in #1? Is 'with' in #1 the same as 'with' in #2 or #3?
Q1: no difference. The "with me" is implied -- how could you take him home with someone else?
Q2: #1 and #2 are basically the same: close proximity at the time.
#3 is a little more general, since they may not actually be physically nearby at all times.