The largest problem with Anakin building C3PO is the fact that in the OT we see multiple droids that are the same make as C3PO.

Which means that either Anakin Skywalker designed an entire type of droid that would be the pinnacle of protocol droids for decades, or that he was exaggerating when he said he built it, and in fact only put it together from spare parts of other protocol droids. That were conveniently lying around Tatooine and that were the pinnacle of protocol droids for decades.

Either way it makes absolutely no sense at all. The way in which Vader doesn't regard him once in the slightest in the OT only compounds the ever increasing validity to the argument that Lucas can't write.

If you watch the series in numerical order there are numerous problems with the story. The largest problems however would be in regards to narrative, and the fact that the series biggest plot turns hold absolutely no weight to them at all. This is most apparent in ESB, with the discovery of Yoda and Vaders revelation to Luke.