No. You need to be an admin to create a project.
There's several workarounds - I wrote addons that created projects as part of a "JIRA Project management workflow" (I'd do that with the script runner nowadays) for example. I'd recommend you take a look through the marketplace too - there's at least one addon which directly addresses this need, and I suspect there's more than one that could help you with it!
Boards do not live "within" a project - they're independent of projects and can include one or many projects. Depends on the filter behind them (project = X is a common one, but you could do "project in (x, y, or z) and issuetype = bug and colour = indigo and and or and or...) Anyone can create an Agile board as long as they have the global permission for "Create shared object". The boards do respect individual project permissions too - if I created a board filter that included "all issues in every project", but I don't have permissions on project "penguin", then it won't show me issues from "penguin".
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