Let me list the frequency of each expression in COCA:
- open a discussion: 9
- start a discussion: 12
- begin a discussion: 19
- open an argument: 0
- start an argument: 24
- begin an argument: 1
It's seems reasonable to me that to open an argument should not be an idiomatic collocation, unlike to start/begin an argument, because an argument doesn't exist until you start it, and therefore there's nothing to open (before it's started).
On the other hand, a discussion about a particular subject exists potentially (in the sense that there already exist some untold facts about the subject of the discussion), and therefore you can open it.
I don't have a definite idea as to why you cannot begin an argument, whereas starting an argument would be fine (take a look at this n-gram too). Surprisingly, start a fight, start a feud, etc are all acceptable collocations, but if you substitute begin for start, they turn out to be uncommon. This merits a whole other question, but maybe this synonym discussion from Merriam-Webster Dictionary could provide a clue, (where it says that start applies especially to first steps):
Begin, start , and commence are often interchangeable.
Begin, opposed to end, is the most general: begin a trip, began dancing.
Start, opposed to stop, applies especially to first actions, steps, or stages: the work started slowly.
Commence can be more formal or bookish than begin or start: commence firing, commenced a conversation.
And to answer your final question about their formality, you can see from the discussion above that by replacing start/begin with commence you could have a more formal or bookish expression. But I personally don't see why you should do that, as there's nothing especially informal about the original expressions.
However, here'e Cambridge Dictionary stating that begin is more formal than start (only when they can be used interchangeably, of course).