What can I do about outlets with no ground and reversed polarity?
Well the most obvious and always recommended answer is to upgrade the old electrical wiring. Very old wires can have insulation falling off or crumbling if you try to start working with it. For a very old installation that may also involve replacing the electrical distribution box (panel) as well. 

Now with that said there is not anything inherently unsafe with a two wire system as long as common sense is used. That common sense has to include:


Not using electrical appliances when the user could be providing an alternate path to GND for the AC electrical power. I.E. do not stand in a full bathtub and drop a plugged in hair dryer into the tub).
Use double insulated electrical tools if at all possible.
Do not use electrical appliances that have frayed power cords.
Replacing the two wire outlets with a GFCI that can trip if the  hot wire current is not equal to the neutral wire current. GFCIs used in this configuration need to be labeled as not providing a safety GND connection.
Consider testing the existing wiring (or get an electrician to do it) to see if the old neutral wiring creates a voltage drop of more than a couple of volts when circuits are loaded. If higher voltages are seen then use of bare metal appliances such as a desktop computer chassis may expose you to the voltage drop of the existing wiring. If this is the case the only real solution for safety is to get rewiring done. 


For the outlet with the reversed polarity the first thing that should be done is to replace it with one wired to the correct polarity. Another thing to look at is to replace outlets with the newer style where the slot for the neutral prong is larger than the hot wire prong.