1. THe obvious one. Speaking English. Book accounts for it, movie doesn't. Doesn't even wonder why he can understand the apes.
A shortcut taken by the writers. They substituted a wounded throat for Taylor having to learn a language. Nothing new. John Carter went with a "magic elixir" in place of teaching Carter the Barsoomian language. It happens in film, get over it.

2. Three military men. Suffer a long hike through the desert. So what do they do? All three get naked and jump into a lake. Leave weapons and food unguarded. Um, did it occur to soldiers/officers that maybe, just maybe, they should let one or two get in water and at least one guard their stuff.
They went from thinking that they would die from dehydration to finding a lake of fresh water. I'd give them a break on this.

3. They see the humans. Wow, lets gets up in the middle of them. You'd think soldiers would know to hide and observe from a distance.
The humans stole their clothes. They chased after them. Then they saw that they seemed harmless. Nothing wrong here.

4. Anyone notice the plastic tube used in transfusion? How the hell do the Apes make plastic? Does that seem like they had the technology to create it?
Dude, they had rifles and photography. I think they could swing plastic or a glass equivalent.

5. They hate humans. Humans are dumb. They like shooting them and putting them in cages. Maybe I'll try to show how smart I am and that I can talk. Yeah, they will want to be friends with me if I do.
He chose his ape counterpart carefully. He saw that Zira would be someone compassionate to his plight. He made the right call. If Landon and Dodge hadn't been discovered as being intelligent, Taylor and Zira could've stayed under the radar.