ericrm, yes it can.
Copper when heated or trying to melt oxidizes very easily forming copper oxides, but can be melted if oxygen is removed from the melting environment, like adding charcoal or flour to the melt or using some other carbon source, this will take oxygen that would otherwise oxidize the copper, and make CO2 gas, when brazing copper brazing or silver solder rod will contain phosphorous which removes oxygen, copper melts at 1084 degree's C, this is much higher than a normal fire, and would usually need oxygen source to reach these temperature's, you can melt copper but it will cost you much in fuel, probably more than copper could be sold for.
Copper salts would be another issue as far as melting them; you would have acidic gasses generated in the melt, and would most likely end up with copper oxides, depending on your starting materials.
As far as electrolysis, unless the copper was very pure it will not plate to cathode as a solid mass but come out of solution as a fluffy moss.
lets look at a few things we may choose to do with some of the copper solutions we generate, copper II chloride we can reuse in our acid peroxide solutions to dissolve more copper, or we can let it go to copper I chloride powder and use this to dry and store to be regenerated later to CU2Cl, or we can use it to remove gold or silver from solutions (usually in recovery processes), or copper chloride solutions will dissolve metals higher copper oxidizes easily when trying to melt it, using a carbon source like charcoal, flour, in series than copper, which would displace copper from solution and make a chloride solution of that metal, copper sulfate could be made from the copper chloride solutions by adding sulfuric acid and heating to a copper sulfate salt, this copper sulfate can be used in electrolysis cells,
now say we had copper nitrate solution left after cementing silver from a silver nitrate solution, we could make copper chloride from it or copper sulfate, we can crystallize the copper nitrate to salt’s had heat them to 256 deg C and drive off NO2 fumes which if bubbled into some water with a little H2O2 would make nitric acid, or with the copper nitrate salts we could separate the copper electrically from the nitrates, using a graphite anode (inert) and make nitric acid and copper powders, we also can dissolve metals higher in series than our copper in this nitrate solution,
The list can go on, of things we can use, from what we could normally consider to be waste.
Solutions take a lot of room, many of these compounds can be evaporated to salts and saved dry, and used later, who knows maybe some could be sold to eBay, or maybe even the chemistry dept at your local high school has use for some of the salts like copper nitrate, or sulfate for experiments, and maybe the teacher would like you to do a demonstration of dissolving silver, cementing with copper and melting the silver for the students, this would be fun to teach the kids, and then maybe the teacher had a few salts he could give you that his lab no longer needed.