This is something that you need to discuss further with your doctor, but they would typically take these images for things like:
Enlargement of specific organs
Kidney or gallbladder stones
perforation of the intestines
Abnormal fluid & air accumulation
Unless your condition was very obvious, a plain radiographic image is unlikely sufficient for a definitive diagnosis. The ER doctor sent you home because you are not dying, and that is really their job, if you are still really sick, you need to check in to see a GP, and have him do some further testing and/or refer you to a GI doctor.
Hopefully your GP will do a complete history, fecal test and repeat the blood work, or ensure that he has a copy of both the blood work and x-rays. who knows, but down the road, it is possible that you could be getting a urea breath test, endoscope, colonoscopy, or CT or MRI. these might be necessary for the doctors to figure things out.
To specifically answer your questions:
not all ulcers will be found on a normal x ray. Sometimes the doctors will have you ingest a substance called barium, and then take an x ray or x ray video (fluoroscopy) of your intestines.
not all cancers can be found on a normal x ray. only certain types and larger tumors will show on an x ray. A CT or MRI would give a better finding.
I work with medical imaging