The border between Morocco and Algeria might be open, but don't build your plans on it. It has been closed for a long time, and I have a feeling this is some sort of political bargaining chip "we'll open it if you do this" kind of issue. From what I've heard, this is not the first time Algeria promised to open that border in the near future.
Algeria is probably safe from turmoil. They're a huge oil exporter, and they went through a decade of terrorism/civil war in the 1990's, so I don't think they will have a full-blown civil war like Syria or Libya. There have been riots, but the government was able to appease them enough.
I don't know how safe it is to travel through it. But then again you biked through Sudan so you should be fine. My friends have stories of bandits on the roads between the major stories, but I don't know how much is hearsay and how much is true. Anyways the situation is much better than Sudan. I think there's trains that work too. I never traveled around Algeria (I only went to Algiers) but the country is developed enough for internal travel.
All in all don't worry about Algeria. It's not a fun place to visit, but it's not really a dangerous one either. You went through Sudan, you should do fine. Tunisia should be even nicer. I've never been there, but I'm pretty sure they're dying to see the tourists come back, since their economy depends on it.
Out of curiosity, where do you come from ? Why do you want to go through these countries ?