Qs

I thought they could represent the presence of Hamlet's Father...

First, Hamlet's father begs Hamlet not to forget him, and to extract vengeance on his uncle. The second thing is that ghosts are usually described with a cold air and a creepy glow about their embodiment. Third, the gloves are the only thing separating from the burning sword. There are a few instances in the play when Hamlet doubts himself, but decides to proceed to plot his uncle's murder because of his desire to save his father from his suffering. Later in the play, when Hamlet threatens his mother, his father appears to him and asks her to be gentle and kind to her, which quells his rage from hurting her.

I don't think I would like to touch a burning sword either... As for whether or not the gloves are personalized when you draw the sword, I do not know.