The Scripture says that the Second Temple was built on the foundation of Solomon's Temple, but was Herod's Temple built in the same place or was it built next to it? (Do we have any records that say so?)
Herod did not build a new temple. The second Temple was standing when he came into power, and Herod did a major renovation on the existing second Temple. He upgraded and beautified the structure, while not moving it at all and leaving the Temple intact. Because of the magnificence of the renovation, the post-renovation second Temple is sometimes called "Herod's Temple".
Both the first and the second Temple were built in the same location on the Temple Mount. The location is mentioned in the Bible and specified in the Oral Law. The Temple can only be built there, and the Torah mandates only one Temple in Israel (hence the capitalization of Temple.)
Herod was known as a tremendous builder or renovator. According to many, this was trying to compensate for his own doubts about the validity of his monarchy. See this link:
He renovated and extended the second Temple. He didn't build it from scratch. There is a concept in halacha that the one who completes an action is considered to be the one who did it. (The idea of the final strike of the hammer) It is only in that sense that Herod could be called the 'Builder of the Temple'. See the following link for details:
Since Jerusalem was the capital of the nation, he also had a palace there. It's location was at the western wall of the city in what is today called the Armenian Quarter. It is possible that if Christian tradition is referring to something 'next to the second Temple', they are referring to Herod's Palace, not the Temple itself. See the following links:
But he also had secondary royal palaces at Herodian, which is southeast of Jerusalem on the way to the Dead Sea and Masada near the Dead Sea (a summer palace on the southern end of the compound) and Caesarea Maritima.
Ultimately, Herod the Great was buried at Herodian.