Hi Victoria,
In my case I went to regular shoes right after the moon boot was dismissed, which was about 8 weeks after my ankle injury (they did not cut me open).
But I wore a sturdy ankle brace in shoes whenever I left the house for work, shopping, etc. (for protection to prevent re-injury-- that's really just cheap insurance and peace of mind). However, once your ankle is mostly healed a brace should be shunned for the most part-- it is after all a crutch.
My view on turning to comfortable but no-support footwear like flip-flops, sandals, and flats for recovering ankle patients is this: you do this at your own risk, and don't feel sorry for yourself if your ankle gets in another mishap accordingly.
We all want our lives back as soon as possible and a return to our familiar routines without delay. But sometimes traumatized ankles have other ideas.
Eventually, if your ankle heals completely, and if you keep doing ankle exercises to keep it strong, aligned, and resilient then I would say by all means enjoy the freedom and comfort of minimal footwear (and especially walking bare feet in grass or sand). If you are still struggling to get your old ankle back then I would say NO, not for now.
As far as persistent swelling goes-- this is a problem that vexes many patients in recovery-- sometimes for months and even into years in worst cases.
Patients have shared on this board many approaches to mitigating swelling. There are really too many to mention here but some that I like are: icing (one patient even bought a cold therapy machine to do it industrial-strength style), elevation whenever possible, compression socks, calf ankle pumping (helps to flush stagnant and dirty blood and lymph from swollen tissues), and hot/cold alternating compresses to stimulate circulation down there.
Edema, or swelling, is essential in early stages of injury to protect tissue and rush immune cells to damaged areas to begin healing. However, tenaciously persistent swelling long after injury is counterproductive. There are probably as many causes for chronic ankle swelling as there are patients.
Best wishes to you.