Can I infer from the fact I am thinking that I have existed for a finite period of time (as opposed to simply "I exist"), irrespective of how short that period of time might be?

It depends. Descartes would say no. The strength of his famous argument "Cogito ergo sum" (I think, therefore I am) is its "performativity", that is, its power to realize what is stated by the simple fact of stating it. But this performative aspect is lost if we declinate the same sentence in a past form. "I think therefore I have existed sometime before now" is a completely different inference. If I say "I apologize" I'm apologizing, while if I say "I have apologized yesterday" I'm just describing a fact about my past life, whose truth-conditions depend on many variables. Whereas G.E. Moore would say that this is commonsensical. In his essay "A Defense of Commonsense" , he writes: "There exists at present a living human body, which is my body.This body was born at a certain time in the past, and has existedcontinuously ever since, though not without undergoing changes; it was,for instance, much smaller when it was born, and for some timeafterwards, than it is now" These are for him truisms,...