Is it possible to be hyperthyroid with hypothyroid symptoms?

Yes. While some symptoms of hypothyroidism are the opposite of hyperthyroidism, some are the same! Both hypo and hyperthyroidism can cause fatigue, weight gain, and poor concentration. Or the symptoms could be unrelated to the thyroid. See your doctor to find out what caused the hyperthyroidism, make a plan for treating it, and then, if symptoms are still present, finding what else might be going on.
Yes. Thyroid symptoms are overlapping and non-specific. A lot of other medical problems can also cause similar symptoms. That's why it's important to be thoroughly evaluated by a physician (history/physical exam/labs/possible imaging) rather than focusing only on symptoms.
Possible. This is not a common observation, nonetheless, it has been described in the literature. Essentially what is happening is that there is an initial transient elevation of thyroid hormone levels due to autoimmune thyroid destruction followed by drop in thyroid production. In medical terms, this phenomenon is called "hashitoxicosis".
Yes. Especially if you have autoimmune thyroid problems this may happen.
Autoimmune thyroid? Some people have autoimmune thryoid & they may oscillate between symptoms of both low & high thyroid levels. You need to have your levels checked by your doc. Check tsh, free t3&t4, total t3&t4, tpo antibodies, tg antibodies.