the guidebook "Mauretani au GPS" , Editions TAKLA MAKANE
, describes the itineraries you asked for.
And yes, no particular issue to drive them with a well equiped 4x4 (at least 2 vehicles for safety).
>I ask myself, how the Algeriens will act if I arrive from the south?
Entry in Algeria at either Bordj Mokhtar, Tin Zaouatine or Assamaka/Ain Guezzam is possible and nothing unusual for Algerians.
I found entry and exit at Bordj Mokhtar very relaxed. It is nothing particularly difficult nor exciting. You should have your visa though.
From this spring experience I know that visa for Algeria were issued in Niamey and Agadez. No experience with Gao. But British travellers who applied for the visa in Bamako were refused the visa despite the letter of invitation.