Is it normal for my teeth to be able to move together and away slightly?

How slightly? Teeth are not like fence poles set in concrete. There is some very slight movement normally. Whether or not you in particular have any dental issues would need to be determined by a dentist who can actually examine you and see x-rays of the teeth in question.
No. Even though our occlusion (the way teeth fit together) is dynamic, i.e. Changing to adapt to the environment, it should not be back and forth. If the teeth move together and away back and forth, this is a sign of unstable occlusion, protrusive interference, or fremitus upon swallowing, or even tongue thrust problem. Consult with your dentist for definitive diagnosis and treatment.
Not normal. Tooth movement as you describe is not normal. If many/most of your teeth are moving, gum disease (periodontitis) is most likely the cause.Over time this causes bone loss which will make the teeth loose. If it is just one tooth that is moving, the cause may be an abscess (infection under tooth) or a fractured (broken) tooth. See your dds to determine the cause, because all are treated differently.
Yes. Although your teeth are in bone, they are not set in stone. It is normal to have a slight mobitity in your teetth.