I think maybe you need a new tip. Sounds like there are "dead" spots on the irons tip that are not transfering heat well. Soldering tips are plated in iron, then tinned with a coat of solder. If you sand the tip, you will remove the iron plating, and once the bare copper underneath comes in contact with solder alloy and flux, it will begin to oxide and corrode away almost instantly. Solder should melt instantly when touched to a soldering iron tip. Along with a damp sponge, you may want to get a tin of what's called tip cleaner. Its a very small metal can of compound that will clean and retin your iron for you when you wipe it on it. It's kinda expensive, but it works. Another thing, when using a sponge wipe the iron just before you go to solder, then when done, put a small amount of solder on the tip before returning the iron to the stand. A cheap way to get a life-time supply of tip sponges is to buy a Cellulose sponge used in wall papering, and slice off small 2" x 2" x 1/2" bits.
the lifted trace can probably be used, just solder to it then reattach it to the board with "crazy-glue" once cool.