Does Cyber Phoenix negate Negate Attack? Normally I'd think so, since Negate Attack targets 1 monster, but Negate Attack is also spell speed 3. Does it's superior spell speed make a difference?

Spell Speed has nothing to do with it being negated. Cyber Phoenix has a Continuous Effect that will be active "as long as" he is in face-up Attack Position, so if any monsters that is a Machine-type Monster attacks, and Negate Attack is activated, Cyber Phoenix will negate the effect.

If it is a monster "other than" Machine-type that attacks first, and your opponent responds with Negate Attack, the Battle Phase will end since Cyber Phoenix can only negate effects that target a Machine Monster.

Simply put...yes...Cyber Phoenix CAN negate the effect of Negate Attack whether it or another Machine-Type monster attacks. This includes the effect of ending the Battle Phase.

It's not simply put. It depends on which monster is attacking "first" versus non Machine type and Machine type.

If you have Cyber Phoenix, Cyber Dragon, and D.D. Assailant on your side of the field, and you attack with D.D.A first, and your opponent activates Negate Attack, the WHOLE Battle Phase ends, and no further attacks may be declared.

If you choose to attack with Cyber Dragon or Cyber Phoenix first, then Negate Attack will be negated by Cyber Phoenix, and the declared attack will continue.

Yeah...note that I said..

"...whether it or another Machine-Type monster attacks..."

...though I should add "...while Cyber Phoenix is in face-up Attack Position on your field."

The difference between both of our post is that you are ONLY addressing a player controlled field of Machine type monsters, while I am addressing a mixed field of Machine and NON Machine, which is why I said it is only simple if you ONLY have one type of monster.