There are several considerations in this situation.
It is true that inadequate sleep could contribute to many of these symptoms. It is obvious that an absence of sleep can create problems, but the quality of sleep is also important. There are different stages of sleep, and if the deeper stages of sleep do not occur, then it can cause symptoms.
Altered sleep can be a primary problem, or it can be due to other conditions that affect the sleep. If your children are interfering with sleep with a sufficient frequency, then that may be enough to cause symptoms, although it usually would require more than just once or twice per night. There are several other conditions in which sleep disturbance can occur. The most common is a condition called sleep apnea, in which a person is unable to get into deep sleep because of pauses in breathing, either because of a lack of trigger from the brain or, more commonly, from obstruction to breathing. Since unable to get good sleep, these patients typically complain of excessive drowsiness during the day, and can also cause headaches and muscle aches. Another condition is fibromyalgia, in which poor sleep appears to contribute to the development of diffuse muscle pain and tender points. Even acid reflux may get worse at night and interfere with getting adequate sleep.
It is also true that psychologic factors can contribute to these symptoms. Persons with anxiety or depression will frequently have either excessive or inadequate sleep, fatigue, and an exacerbation of a chronic pain syndrome, including headaches or muscle aches. Therefore, your putting it down to stress could certainly be true, and if a medicine directed towards this relieves the symptoms, then that would confirm that it is contributing to the problem. It is also true, though, that certain antidepressants have been shown to help many of these other conditions, including decreasing the frequency of migraine headaches and easing the symptoms of fibromyalgia, and may help by a variety of mechanisms.