Can you FORGE aluminum in a REGULAR WOOD BURNING FIRE?
Yes, but you would need an ample air source to feed the combustion process of the forge. Aluminum melts at about 1200-1400 hundred degrees Farenheit, most pot metals will melt at around 750 -900 degrees. In the olden days you saw blacksmiths pumping a Bellows to provide the combustion air because they needed to get the metal Red hot so as to work and stretch the metal. Aluminum with its high melting point required constant heating and air being applied to get it to that required heat for working the metal. In your case an air compressor supplying a high level of air mixed with oxygen would be the only way you could do this unless you had a high temperature metal forge unit. The air bellows pump is built into the forge.
You need to make sure the damper is open. after arranging the wood you will be burning, from small to larger pieces on top (and possibly some paper), hold a lit piece of paper up near the flue. This will get a draft started so the smoke doesn't come back into the house. When you see the smoke from the paper going up the chimney, light the wood. Do not use flammable liquids to start the fire.
aluminum is a tricky metal, too much heat will burn it it gets pitted and losses all its strength a bellows would help get the fire hot