Does an oath or vow take effect if one makes an exception by saying "Insha'Allah (God willing)"? And does an oath take effect concerning something one believes to be true based on their predominant assumption?

If someone making an oath says "Inshallah" (if Allah wills it), then there is no expiation required of him. As for conditioning a vow upon Allah's will, scholars have differed on this matter. The majority opinion is that the vowed act is not binding when such an exception is made, whereas the Maliki school holds that fulfilling it is obligatory. Swearing an oath based on what one largely believes to be true, and then it turns out to be otherwise, falls under the category of a 'vain oath' (laghw al-yamin (definition)), which entails no consequences.