I see that tag synonyms(aliases) exist, however, I can't seem to figure out how they actually get created. There's at least one question on meta where synonyms are proposed ... Is that something that only moderators can do, or can other users do that sort of thing?
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When you go to a tag there is a link for synonyms.
When this link is clicked, you will see the following message:
Right below this message you will be able to suggest the synonym if you meet the requirements.
Yes they can. See this page from the privilege section: link. Since the requirement is high, low rep user usually just post a request on meta so that others may do the work.
You also have the link in your privilege screen.
Click here to check.
Edit: forgot the free-hand circle ^_^'