Yes, you may do that. The field names are Start Date and Due Date.
You could set those fields manually by putting them on your edit views, or update them automatically with automation rules, workflow, or scripting. For some users, they cascade the values from the parent down to the children, and for others they do the opposite, rolling them up to the parent epic.
Yes, and you would need Jira Cloud Premier for that...which is a more expensive license level. Please look here for details to see if that might meet your needs:
As an alternative, there may be marketplace add-ons which do this at lower cost, particularly if you do not plan to use the other features of Premier.
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