Hi, my sister Beth.
I didn’t respond to your question the moment I saw it, because it’s really a tricky one. I’ve been thinking about it since then, but it seems I still don’t know an exact answer to it. However, I can give you some general ideas.
First, the expressions you’ve listed have the exact same meaning as 'again, another time, etc'. Except that there is no such expression as又次, so you may just forget about it.
I think the essence of these expressions are about the differences between 再and 又. You know ‘goodbye’ in Chinese is 再见, so you can see from it that at some point再indicates ‘departure’. So, if you’re talking about something which are about to happen, generally you should use再. The most simple example would be:再见others are:希望我们可以再见面。/具体问题我们见面之后再谈。/什么时候可以再见到你?/那家餐厅很不错,我想改天再去一次。
又can be used when it is happening or has happened. As: 你好,我们又见面了。/经理刚才又过来了一趟。
Other than ‘again’, 又can also be used as ‘and’, eg. Beth的孩子聪明又可爱。
Some lively examples:
Mother: 你怎么又跟人家打架了?(it is happening or has happened,又)
Son: I’m so sorry.
Mother: 你要是再敢跟人家打架,看我怎么收拾你!(an hypothesis into the future再)
There is a very important feature of又I think you should know is that sometimes又 is more just about ‘again’. When using it, it shows the speaker’s negative emotion, like furiousness, sadness, anger, etc. Like the example I gave you:你怎么又跟人家打架了?/他考试又没通过。/他又感冒了。
There are just some general ideas as I understand the two words, however, exceptions are possible.
As for 再次、再一次、又一次, as you can see from the number of the words, they are relatively less casual than just using再or又, often seen in writings or heard in presentations.
About再、又、再次、再一次、又一次, if I keep typing like this, it would be a linguistics term paper. I think we can discuss more on skype, don’t you think? You may make up sentences with the word 'again’, and I’ll tell you which expressions among them are the most possible choices. I think it’s the best way to understand them.