Voicemail is completely reliant on your cell phone carrier, you must talk to them about fixing it. Its cutting out on people or hanging up on them, since the voicemail is not working. It also might be the problem behind how it hangs up immediately, it also could be a phone problem. But Im pretty sure its your cell phone carrier. For example I had a text message problem on my old smart phone, youd expect it to be a problem with the phone, but instead I had to get it re-setup through by cell phone provider as the settings messed up. (Settings sent to phone from the cell phone provider, not in the phone itself/originally)
This has nothing to do you with your phone, and everything to do with your cellphone provider. Give them a call, check their website, or your bill for a customer support number.