No. Port is a 'fortified' wine, with much more alcohol and much stronger aromatic flavouring than either burgundy or any other conventional wine. Do not substitute either port or sherry for conventional wine in any recipe.
In California there are no laws that regulate which grape varietal can be made into "burgundy". Californian "burgundy" is typically red. In France, however, Burgundy is a region. Wines that come from...
Burgundy is a region in France that produces both red and white. Commonly when someone says "Burgundy" they are referring to the red wine made with Pinot Noir grapes. That would be a good red-wine...
Depends on what you are cooking. In my cooking, I've found them to be interchanghable. Red cooking wine is NOT a substitute for wine meant to be consumed as wine -- drinking it.