Could we estimate the age of the universe based on the planar property of the Solar System? The Big Bang scattered planets and stars everywhere in three dimensions.

But after billion years of moving and interacting with each others through gravity, planets moved on the same plane.

Given the very small but non zero friction in space, could we give a good estimate of the age of the universe?
First, the Big Bang did not happen at a point. I cannot emphasize this enough. It happened everywhere in space at the same time. You can't think of it as an explosion from one location pushing everything else away.

Furthermore, planets and stars didn't exist at the time of the Big Bang. The Solar System, for example, formed about 4.5 billion years ago, while the oldest stars - Population III stars - formed within a few million years after the Big Bang. At the beginning of the universe, everything was in an exotic soup-like state: a quark-gluon plasma.

Therefore, your hypothesis is flawed in two places.