I would agree with your analysis of the first sentence.
The post-modifier, left, represents a reduced relative clause.I have 10 minutes (which is) left.
In the second sentence, the verb, have, is used as a causative verb.I have so much fun planned.
The sentence would mean 'I cause so much fun to be planned.'
I have my hair cut.
'have' here is also a causative verb.
Student or Learner