I just wanted to be sure about the position of the conjunctions. Could you please check these sentences? Thank you.

1) He fears they will have an accident, injure themselves or that their motorbikes could break down.
2) While I was getting ready for school, my sister had a shower (is "had a shower" possible??)/ my mum phoned me.
3) After I had got ready, I left home.
4) I don't belong to any charities. If I donated money, I would donate it to UNICEF
5) He fears they will/might/could/ have problems with their motorbikes.
Can you say "their motorbikes will have problems??"

1) He fears they will have an accident, injure themselves or that their motorbikes could break down.

There's a major problem here - non-parallel construction. Try this: He fears they will have an accident, injure themselves, or suffer broken-down motorbikes. In your series, the elements needs to be alike; you had two verb phrases and a noun clause, and I changed it to three verb phrases.
http://grammar.ccc.commnet.edu/grammar/parallelism.htm

2) While I was getting ready for school, my sister had a shower (is "had a shower" possible??)/ my mum phoned me.
"took a shower" is better; everything else is fine.

3) After I had got ready, I left home.
"had got" is awkward; try this: After getting ready, I left home.

4) I don't belong to any charities. If I donated money, I would donate it to UNICEF
People don't "belong" to charities. They volunteer for charities; they donate to charities; they help with charities; etc. I think I wrote this before.

5) He fears they will/might/could/ have problems with their motorbikes.
Can you say "their motorbikes will have problems??"
Your first phrasing is far better; the second phrasing sounds as if the motorbikes are people!!