It's not necessarily your use of the word 'Bother' 'Bothered' etc. There are just some words in the sentences from my point of view which elongate the sentence unnecessarily. The school one could be simplified down to 'I'm not bothering to go to school today for the test'. Apart from grammatical stuff, I'm not seeing too much wrong with the others.

These sentences are all correct. I'm pretty sure that american english speakers dont use this phrase - but in the UK it is very commonly used. Good luck.

I think, there is no sentences like these. but you can say. I'm sorry to bother you but I don't know you're sleeping.