It is called unarmed, after all.

Yes. You'll do far more damage with a Power Fist than with your bare hands.

Boxing tape does just fine for me, but it does do a lot less damage than proper weapons. It's up to you, depends if you want a bit of a challenge. It should get easier as you get more perks for it.

Is a weapon needed? No.

Will you be successful with out a weapon? Not that often.

Boxing Tape, Brass and Spiked Knuckles, Love and Hate are all small unobtrusive weapons for an Unarmed character. Note, boxing tape just fatigues them.

That depends on what you have in mind. If you're going to specialize in unarmed combat, then yes. You'll do a lot more damage with an unarmed weapon. But if you're mainly using guns of any sort as your principle weapon, then no you don't.