When Sera holds Kirk and Singh at gunpoint outside the security door at the facility, she is holding a Sig Sauer handgun. Once inside the facility, the gun is now a Glock, an entirely different brand and look.
In the car, while having a conversation about names, Kirk says "Sam is his middle name. Most people call him George", clearly using present tense when scenes earlier it is explained that Sam is supposed already be dead in his timeline.
Finding a cold fusion reactor in Toronto with a Tritium watch from the 1990s would not work. Tritium is detected via the beta particles (electrons) it emits, which can't even penetrate a sheet of paper.
The painting in Pelia's possession is The Concert by Johannes Vermeer (1665-66). However, this painting was not stolen from the Louvre. It was stolen from the Isabella Stewart Gardner Museum in Boston, MA, in 1990.
Noonien-Singh and Kirk comment on a sunset on Harbourfront in Toronto, but they looking are southeast. So technically it was a sunrise.
Noonien-Singh and Kirk are shown arriving in Vermont, however they wouldn't have been able to cross over the border from Canada because neither one had ID.
This is explained when they are first talking to Pelia and Noonien-Singh clearly states they had to bribe a border guard.
It's stated in Space Seed (1967) that Khan left earth in 1992. However at the beginning of this episode Kirk & Soong say they arrived in the mid 21st century.
When in Khans room and before she departs La'an Noonien Singh leaves the handgun on Khans bedside table. As a security officer she should know the damages and risk of this, especially in editing the time line.