It's a classic, tricky problem. Sometimes it's not clear who a pronoun refers to. But if you spell it out every time, your text sounds very awkward. You can say "she" twenty times in a paragraph and it flows naturally, but say "Monica" twenty times and it sounds repetitious.
I'll ditto ClockeWork on his comments about your second paragraph. There really is very little ambiguity.
How we match pronouns to their antecedents can be a complex thing. A pronoun doesn't always refer to the most recent noun or anything like that. Often we figure it out by our understanding of the people referred to, or logical analysis of the action. It would, I suppose, be possible for someone to read that paragraph and think that Monica's mother went away on a trip every time that MONICA had a fight with her father. Or, someone might start to read the last sentence thinking that when Monica's mother went on a trip, Monica stayed in a hotel. But then when they got to the final clause about coming home, they would probably realize that you meant that it was her mother that stayed in the hotel.
Often we do this without even thinking about it. It is only when we realize that our assumptions don't make sense that we go back and say, "Oh, 'she' here must be referring to the lady in the red dress, not Sally's aunt ..."
Two tips: 1. Go back and re-read a sentence, preferably some time after you've written it, and see if the pronouns are obvious. This should be a part of your proof-reading. If you have someone else available to proof-read, that's better still. Someone else may notice an ambiguity that you didn't realize was there. 2. If someone is identified only by a long description, it can seem particularly awkward to have to repeat this to identify him. Like if you mention "the clerk at the hardware store with red hair" and "the clerk at the hardware store with brown hair", you can be tempted to just say "he" rather than repeat that whole description when "he" is ambiguous. In such cases, it can help to give the character a name, or invent a nickname on the spot.