Question: Almost all of my students use the word "cooperate" in this way: "My company cooperates with X." X can be a company or person. For example, if the student works at MediaTek, they will say "We cooperate with Sony and HTC for mobile chip design."

In looking in several dictionaries, the first definition is always "to work together toward a common goal," however the example sentences do not have "work together" as the feeling but "do something you don't really want to do." For example, "ask the child to cooperate and go to bed." I have heard sentences such as "The suspect is cooperating with authorities," or "Her child is uncooperative in class and causes disruptions."

I always thought cooperate has this feeling of "forced" or "unwillingly obliges." Before coming to Taiwan, I never heard people use "cooperate" in the sense of "work with another company." I usually suggest "We work with HTC and Sony." as a better choice. Also, they will sometimes say "I need to / must cooperate with my customers." This usage seems to support my idea of the word, however when I ask for further details they don't feel "unwilling" to work with customers. Some do, though.

Does cooperate mean to work together without any emotional feelings or willingness or obligation? Is cooperate a neutral word?

Answer: 
The word itself is neutral. It's only the examples you gave that have a context of implied threat.
Cooperation, like assent, can be wholehearted or grudging.

In American English, "work with" is more commonly used to describe mutual ventures, as you suggest: 

**FlimFlam Inc is working with Disney to create a virtual theme park with hundreds of imaginary rides replete with long waiting lines.**