Will anything happen if the Earth were to be just 1 cm further away from the Sun?

It can't. The Earth is like an enormous gyroscope; it cannot tilt more or less without some titanic external force doing it, and the force required would most likely cause catastrophic damage to the planet. The tilt of the Earth does naturally vary slightly over time; this motion is called nutation. It's a fairly complicated motion with many terms; the largest one amounts to a change of about 17 arc-seconds (about 1/200th of a degree) over a period of 18.6 years.