MUST VS HAVE TO

Hello everyone,

I invented these sentences about the difference between MUST and HAVE TO:
(If I wrong to write some of them, please correct my spelling mistakes).

REQUIREMENT, RULE:
You lost the match. For penitence you have to count to 100.
To play in that playground you must be 10.
You must make less of 4 mistakes to pass the exam.

PROHIBITION:
You musn't/must no smoking here (the signs says).
you mustn't smoke on buses.

RECCOMMENDATION:
You must read this book, it's fabulous!

INTERNAL OBLIGATION:
I must tidy up my room. it's a very mess!
I must to go. (it's late for me).

EXTERNAL OBLIGATION:
I have to go (My mom is telling me).
I have to tidy up my room. (My mom is telling me).
if I want to keep living here, I have to pay the bills.
I have to make copies (I'm an employee, not the boss. It's my work/job).
He has to look after his child (It's his job/work).
The workers have to work on Saturday mornings.
This course doesn't suit me. I can attend it because in "those times/that time" I have to work.
I have to do well on my test because I want to go a good school (external reason).

DEDUCTION:
They are Americans, I could hear (I heard accidentally, I listened) their accent.
I'm looking for the keys, I leaved it here. They must be in my room.

You must be wearing hair gel/You must have gel in your hair.
(I'm looking at his hair and I'm sure about that).

In these sentences I don't know which modal verbs I must use.
My DOUBTS:
- He doesn't collaborate anymore, I have to/must repeat the same things twenty times/several times! (External obligation?).

- I must read four books to prepare the exam. (I'm a student but the teacher doesn't obligate me to read them but it's important).

- You must/have to wait me before crossing (conversation between me and my child/ my child and I). (Reccomandation or External obligation?).

- I must go to the gym this week (I attend this course and I pay for that, but no one obligates me to go but it's important to me).

- I have to carry your bag because the luggage is broken. (External reason)

- Joseph has stomacache. He must make a poo (Advice, suggestion).
- Joseph has stomacache, he has to make a poo (External reason).

- After eating too much candy, Joseph has been got sick/ has been felt ill.
- He has to see a doctor. (External reason).
- He must see a doctor. (Recommendation and deduction).

- Don't eat the row meat! I have to/must cook it. (External reason or requirement?).

- Could you move please? I have to pass. (you block me the road, external reason).

- I have to/must change my clothes/get dressed.
(Someone threw a drink on me. External reason or evidence?).

I found the following sentence on the internet and it's true, why?

You've got to see that movie, my friend.

(Why is it use "have to"? I suppose it's an external obligation: it's beautiful).

You must see that movie, my friend (reccomendation).

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Thanks a lot!