Some Harry Potter 'questions'

There are plot holes so large you could drive a truck through them. Here are a few in 'question format':

• If Fudge would not believe Harry in the Goblet of Fire that Voldermort was back, why not just look into his memory (like Harry and Dumbledore do on many occasions to many different people's memories)?
• Why didn't Harry just kill Voldermort on the many occasions when he had the chance? Harry knew what the score was and that nothing would end till one of the them was dead.
• When Harry and Dumbledore apperate back at the castle after Dumbledore is weakened and they heard others coming (and Dumbledore seemed to already know what was going to happen)... why didn't they apperate back out?
• That bothers me too. Dumbledore always acted as if his death was apart of a plan and you never really knew if he died or not cause of the way he kept talking. In the end he did die and for no apparent purpose or plan. It was kinda dumb.
• If the luck potion and the timeturner were both so powerful why weren't they used at REALLY critical moments of Voldermort's rise? For that matter why didn't Dumbledore keep the Sorcerer's stone (or a potion of it) too?

I understand these were 'kids' books written by an a first time writer but there were some big holes there...