You should check the standard reduction potentials of each reaction in the cathode and anode. In a fuel cell (which is a type of galvanic cell), the hydrogen oxidation reaction occurs on the anode at 0V vs RHE while the oxygen reduction reaction on the cathode occurs at around 1.23V vs RHE. As you can see, the anode reaction occurs at a negative potential compared to the cathode reaction. That is why the anode is marked negative and the cathode is positive.
At the anode, hydrogen is oxidised (losing electrons). My first
question is this: why is the anode negative if it is attracting
electrons?
The anode is negative because it is accumulating electrons from the oxidation of hydrogen atoms.
My second question refers to the fact that once the hydrogen atoms are
iodised, the H+ ions move through the electrolyte towards the oxygen
ions at the cathode, reacting to form water. If the anode is negative,
why would positive H+ ions move away from it?
As there is an accumulation of electrons at the anode, electrons repel each other and are attracted towards the cathode where there is a deficiency of electrons. At the same time, the cathode is becoming the "meeting place" for the incoming electrons and provided oxygen atoms which attract the hydrogen ions to drive the oxygen reduction reaction that forms water. So H+ ions are attracted to the cathode due to the increasing number of electrons in the cathode.