Should I use normalize.css in my Bootstrap project?

No, you shouldn't.

A modified version of normalize.css is already included in Bootstrap 2.3.2. Here it is.

UPDATE: the docs makes this even more explicit, and now they use the original code, not their fork.

UPDATE: (Bootstrap 4)

For improved cross-browser rendering, we use Reboot to correct inconsistencies across browsers and devices while providing slightly more opinionated resets to common HTML elements.

Source: https://getbootstrap.com/docs/4.1/getting-started/introduction/#reboot