Can mattress cause neck pain?

A mattress can cause neck pain. This is probably due to the mattress not providing enough support for where the persons head rests because the mattress is to firm, so the mattress is too hard for a person to comfortably rest their head on. Also, if the mattress is too soft for where your head rests it will make your head drop to far into the mattress hurting your neck because of the positioning of the neck in comparison to the spinal cord which should be in a natural position if the rest of the mattress supports your body just but has signs of indents and sinkage around where a person would lay their head upon. 

Can a bad mattress cause neck pain and headaches?

Yes, they can but it also maybe the pillow you are using as well. Headaches can be caused by neck pain, which probably will affect the quality of sleep leading to headaches due to lack of sleep or deep sleep. 

New mattress causing neck pain

This is probably because the new mattress is too firm and has not been broken in, so you are not able to let your neck sink enough to be comfortable or close enough to a natural position for your head to rest in. Moreover, if a new mattress is causing neck pain it just means that the quality of the mattress was not good enough. A good mattress would be ergonomically designed to avoid issues such as neck pain.