Why doesn't Macbeth put on his armor before he goes into battle?
Macbeth doesn't put on his armour before battle because the witches have told him that "none of woman born" can harm him. He therefore thinks he is invincible - and why would he need armour?
Your second question is more difficult. We never know precisely what Macduff thinks - the only clues come in Act 2, Scene 4. He comments, when asked who is supposed to have killed him, "Those whom Macbeth has slain". And he also comments that Malcolm and Donalbain's escape "puts upon them suspicion of the deed". I'd say he suspects Macbeth killed Duncan, and that Malcolm and Donalbain, and the servants, are innocent.
Why does he order the murder of Banquo and Fleance? Because the witches have told him that Banquo's sons will be kings, adn he wants to secure his throne. So, it is because Macbeth's throne is weak that he resorts to more murdering.
Hope it helps!