Yes fire is the weakness, and for a weapon I might have to go with the lance or longsword. The lance can hit the gut, head and tail of the creature easily while longswords decimate the creature before it can do much. But lances have the bonus of blocking so go lance.
1- Go with whatever weapon you have with the highest raw damage.
On a monster like Gigginox, elemental damage probably won't prevent a time-out, and probably won't make it noticibly easier. However, if your strongest raw damage weapon has fire element, then it's a win-win situation.
2- Lance. Unfortunately, there is no lance with fire attribute attainable before fighting Gigginox as far as I know.
There is nothing either good or bad, but thinking makes it so. William Shakespeare