Translation:Can you help me to throw away the trash?
Throw away the trash and throw the trash away should both be accepted.
This is not natural English. To throw something away is to put it in the trashcan. You don't need help with that. What you might need help with is taking the trash out. Like the plastic bag inside the bin needs to go outside into a bigger bin or to be collected. It may well be that "throwing rubbish away" and "taking out the trash" both translate to the same phrase in Chinese. But either way the English should be changed to something English speakers actually say instead of mixing two concepts that are distinct in English.
I think 帮 in this case should be better translated as "for", it's not really help but more like "Can you throw the trash away for me, please"
"can you help me put the rubbish out" ...as in the household chore....was rejected and I'm wondering if it's acceptable or not?
I think ”take out the trash" is more idiomatic and while it'd commonly be implied to mean throwing a bit of rubbish e.g. a food wrapper, if placed in the context of a household chore, it is acceptable.
I'd use "for me" rather than "help me to". “帮” is often translated as a preposition rather than a coverb.
It said "help me to throw away" is an alternative, but "to" is never use between help and another verb.
Help me lift this. Help him read that. Help her do blahblahblah.
"Could you help me" should be acceptable, more so than "Can you help me". It is a standard way to ask an indirect question.