2)And also, why do we write cell notation involving hydrogen ions this way Pt|H(+),H2... ? Why is there Pt as elctrode while not in the half reactions?
The electrode only acts as a pathway for electrons to flow within the electrical circuit. It is not directly involve in the chemical reaction, ie. what's happening inside the aq medium.
3)Why would the electrode that electrons flow to be the cathode(positive)? Should the electrons make it more "negative"?
electron from low potential to high potential. moreoever, electrons are attracted to positive charge. if electrons were to flow towards an electrode, that electrode must be positive.