"invoice"

[tooironic]

I have a couple of questions about the word "invoice" in Chinese.

Firstly, my understanding is that the noun "invoice" is usually expressed as 发票 in Chinese, but I've also come across 发单 and 发货清单. Are there differences between them?

Secondly, as for the verb "to invoice", I've seen it translated as:

1. 寄送发票
   
2. 发货单
   
3. 开发票
   

Would I be right in saying that 1. might translate as "to send...", 2. as "to issue..." and three as "to write..."?

Thanks!