In general, stroke symptoms are worst at onset and then slowly improve. A few exceptions to this are pain syndromes and "phantom limb" syndromes, which often start weeks or months
after the stroke. Therefore, headaches could occur several months after the stroke. I have seen this on a handful of occasions. It makes sense to repeat head imaging, to make sure that no new strokes or anatomic issues have developed. If everything looks ok, best to treat the headaches with medication if they are troublesome to you. Examples of medications that I use frequently to treat headaches include topiramate, amitriptyline, nortriptyline, and gabapentin.
With regards to your final question - any new symptoms should prompt a new image of the brain. Although numbness and/or pain can often come several months after a stroke, and sometimes old stroke symptoms can return in the setting of an underlying illness or fatigue, a truly new symptom should prompt evaluation for a new stroke.