No, you don't! The game is free and should be totally functional on its own - if you get a message telling you to purchase Modern Warfare 2 to have access to everything, that's actually a glitch, as listed on our Warzone 2 error codes page.

Yes. All Modern Warfare II attachments will be available in Modern Warfare III and can also be used on MWIII weapons, assuming they are attachable to the weapon in question.