What is the ruling on one who doubted the number of circumambulations (Tawaf) and completed his 'Umrah, then repeated it from Al-Tan'im, and is a sacrifice (dam) incumbent upon him for that, and how can the sacrifice be offered if he is outside Saudi Arabia?

You erred by undertaking that act without verification and without asking people of knowledge. What was obligatory upon you, at the very least, was to build upon what you were certain of. So, if you doubted the number of circuits of your circumambulation while performing it, you should have based it on certainty, which is the lesser number, according to the majority of jurists. Since you remained in the state of ihram (definition) until you performed the second Umrah (definition), the circumambulation and the striving (sa'i) for the second Umrah are sufficient for the Umrah you did not complete. You were not required to assume ihram from the miqat (definition) because you were still in your first state of ihram. As for the prohibitions of ihram that you committed, those that fall under the category of causing damage (itlaf), it is more cautious to offer a fidyah (definition) (expiation).