I might have both hypo and hyper - thyroidism, is it possible?

Not at same time. A person might be both hyper- and hypothyroid at different times in his/her life, but you can't be 'actively' both at the same time, nosiree.
Depends. Some people with autoimmune thyroid disease can swing over time between being hypo and hyper thyroid, however at any one given minute, you can be one or the other, not both. Certain medication, such as cytomel, (liothyronine) if dosed incorrectly, can also cause someone to swing from having hypo and hyper thyroid symptoms in the same day.