Look around the thing you are working on, including the case. If it says ROHS or if it has Pb with the circle slash thingy, it is for sure lead-free. Lead free is harder to work with and will need a higher temperature.
If your Hakko is like my Hakko, it has a temperature sensor near the tip, and when the temperature drops, it kicks up the power, reducing the need for a bigger soldering iron, and in general making your life easier.
I like the clip the small leads idea, as long as you are SURE that part is bad. Don't pull on the clipped legs too soon or you will lift the copper trace and come up with a new swear word.
If you have ANYTHING else in your house you could give two hoots about, I would try a few unsolders on that first.
I learned a technique at a previous job, they had this tool that is like a heat gun, shoots a concentrated stream of hot air, you point it at the chip and apply a gentle sideways force, when its ready, it moved, this is for chips with small pins though.
That tab on the new FET is going to be a bear to do right, we have a problem at work with voiding under the component, the voiding reduces thermal conductivity and the component over heats.
