If a bronze user gets >500 total submissions, and later downgrades to free, does he keep all the previous submissions?

Can new submissions be added as a free user after downgrading from bronze w >500 total submissions?

I believe you are referring to the total submission storage limit?  If so, if you are over 500 stored submissions and downgrade back to a Free account, the most recent 500 submissions will be kept and the rest will be deleted after 10 days. 

If you need to store more than 500 submissions after downgrading, we do offer a Storage subscription:

https://secure.jotform.com/pricing

You can also delete previous submissions to reduce the stored submissions total.