Hello, I'm not sure how I got notified of this, however, * produces what is called a bullet point. It is used to sign that you have a particular point to make in the particular writing. As for common usage... it can depend which area you are posting in. The most recognisable example is DYK. Bullet points are often used more than anything else in DYK.
Although it may be used numerically more often in votes, the reason for that is, when each post is not indented, which is most commonplace in a list of votes, each bullet point makes scanning and counting through the votes much easier as the eye drifts from bullet point to bullet point to easily see the start of each vote which would otherwise be lost in a wall of indiscriminate looking text. The standard is to use a bullet point accompanied with a word in BOLD text, making each start line very visible and easy to scan.
For discussion purposes, the use of bullet points is easy. It is supposed to be used when making a list of individual points. Often a post will start of with some text introducing the list or other point of discussion, then a list of bulleted points, then some more text to conclude and sign off, and variations on that theme. It's not a Wikipedia theme. It's just adopted by Wikipedia from common formats, i.e. mainly text-book formatting.
Then # produces numbered lists, similar to bulleted points, applications are obvious.
Using (***) or (:::) cannot be uniform in each case because there is a purpose to (***). Forcing one or the other will not improve presentation of text and is unlikely to be accepted, at least on the en.wiki, where most people, although they will use these codes in various ways, will have a good idea of why these tools are useful. It would be like enforcing a spellchecker. You'd come up with no spelling mistakes, but then I couldn't post a word or lettering which is not in the dictionary. How am I supposed to discuss algebra in that case?
In my experience ":::*" is the standard way used to indent with a bullet point (as opposed to "*:::"). The only problem with mixing indentation styles is that posters do not undrstand how to make the length match. If the last post was ":::*" and I indented with "::::" which means another ":" added, the text will not indent. I must add one for the "*" also, and then the text will appear indented.
I may think the reply scheme as linked on the beta bluster is going to be an improvement for several reasons, and support it on strong terms should they be required. However, as someone who types rapidly with about four fingers, I make a lot of errors, which I usually catch as soon as they are made, before posting, but I think I need to be able to edit my text quite easily. I'd like to see the section edit button stay there. I would suggest with both the reply and section edit available in that style, the reply will become the standard used button except for correcting errors. In fact, I'd suggest have an edit button beside the reply button, and a section edit button in this format. I suspect motivation behind removing the edit button may include such things as entering many replies at one time, which can make review of postings difficult. With this new proposed format, I expect that behaviour would become frowned on because it would be much easier to use the reply button.
Sorry for the length, as the old joke goes, I didn't have time to write less.!