My question is that can you continue after running out of lives and freeplay enabled?

So far there aren't any rules against using continues, especially with freeplay. Regardless, the current route is laid out such that continues are not necessary or used in any way (because it's slower to use continues). I'd say it's allowed for now just in case it can be abused to skip certain sections of the game or for practice reasons.

Currently, there are no rules against continues! Generally, you suffer a time loss on getting hit, with exceptions to a couple of points (like the first sequence of the boss in 2-3). Continuing does clear the current area for you but hard to imagine saving time in this regard unless the hits are taken at points where the game moves on during the damage animation.