Is my entering the examination room of a doctor, to whom I swore not to go for a check-up, after I only recognized him upon entering, considered a breach of oath? And in the event that expiation is obligatory, is it better to give charity or to fast, knowing that I have no financial income?

If you knew that the doctor was the one you swore not to go to for treatment and you accepted that, then you have broken your oath and must pay an expiation for breaking an oath. This expiation is feeding ten poor people, or clothing them, or freeing a slave. If you are unable to do any of these, then you must fast for three days; as Allah Almighty says: "Allah will not impose blame upon you for what is unintentional in your oaths, but He will impose blame upon you for what you intended in [breaking] your oaths. The expiation for this is feeding ten poor people from the average of that which you feed your [own] families or clothing them or freeing a slave. But he who finds not [the means to do so] - then a fast of three days [is required]. That is the expiation for your oaths when you have sworn. And guard your oaths."