Civilization – If one player doesn’t own a DLC-civilization, can the others still use it in multiplayer

Unfortunately, I don't believe it's possible to do. The fact that the minimap changes on reload probably has more to do with the way the map itself is handled -- when you start out, the map is centered on your revealed position, but as more of the map becomes revealed, eventually the actual "edges" of the map become visible.

I'm betting Civ V doesn't keep track of how your map started, so on the reload it recenters the map based on the revealed area and not your starting position (which invariably will result in the center of the game map being in the center of the minimap as the greater whole of the world becomes explored)

Your wrote a lot of side questions, but I will answer what seems to be the main one:

Can a player rejoin a Civilization 5 multiplayer game after being disconnected?

In theory, yes. I've personally done this. Try reconnecting by finding your friend on Steam, right clicking on them, and selecting the "join game" menu option, it will take you right back to the game.

However, its quite possible for whatever caused the disconnect in the first place to prevent you from reconnecting. I've personally experienced this as well, due to my internet connection simply dropping. If this is the case, that problem needs to be fixed before you can reconnect.

If you are having problems with a bad connection, there is one more thing I'd suggest you consider. As of now, in multiplayer Civilization 5, you can't just save the game. The only saves you have are the autosaves. So, I'd recommend changing your settings so that you autosave every turn. Then, at worst, you can reload the last save from one turn ago.