Once your residence card expires, you fall into a legal limbo. There’s no simple way to extend the validity of the old card, which is what I presume you mean by “rollover”.
I can’t answer your question about using the health system.
Here are a few threads where folks have shared their experiences traveling with an expired residence card:
There’s also something called a “tacit approval” that may or may not have any real effect while crossing borders:
Be aware that before you receive your new residence card, flying in and out of Schengen from Portugal may be a very different experience than via another Schengen country.