The architect didn’t have his hands cut off:

The head architect of the immense project, Ustad Ahmed Lahauri, went on to have a hand in laying the foundation of the Red Fort in Delhi. Shah Jahan would have probably not commissioned further work from the architect if he was indeed maimed. 

The Taj had to be disguised during World War II:

To escape the searching gaze of passing bombers, the Taj Mahal was covered with an enormous scaffold so it looked like nothing more than a large bamboo stockpile from the air. The same thing was done during the India-Pakistan War in 1971 and following 9/11, except with a green cloth to camouflage it completely. 

The Taj Mahal is taller than the Qutub Minar:

At 73m high, the Taj Mahal is about five feet taller than the perceptively vertical Qutub Minar in Delhi. The Taj is also taller than the Hagia Sophia in Istanbul, but is significantly shorter than other similar shaped monuments such as the St. Paul’s Cathedral, the Temple of the Vedic Planetarium, and St Peter’s Basilica. 

Shah Jahan wanted to build a black Taj Mahal:

Mughal Emperor Shah Jahan wanted to build himself a comparable mausoleum to the one he built for his beloved wife, Mumtaz Mahal. Historians have uncovered that he began work on the other side of the river on a massive, black tomb, but he was deposed by his son Aurangzeb before the project could significantly materialize. 

The Taj was ingeniously designed to guard against the elements:

The four minarets surrounding the main structure of the Taj face slightly outward. This was done intentionally, so that  in the event of a natural disaster like an earthquake the pillars would fall outward, protecting the tomb. The location of the Taj was also inspired as the water from the Yamuna kept the timber foundation strong through the years, preventing its erosion.