The 'mechanism' of length contraction

[Nugatory]

Planck length is defined as the smallest unit of distance.

It is not. It might be the smallest distance that we can measure, but even that is by no means a settled question.

One of the reasons I'm asking this is because I'm trying to connect this to the case of the rotating disc, where each point has a different velocity and therefore a different time dilation value. Since this depends on the distance from centre, and there are infinitely many points, can a distance from the centre be calculated for each point (and therefore the velocity) despite the number of points not being countable?

We can certainly calculate a distance for any single point - write down its coordinates, do some calculation (trivial if we're using $r,\theta$ polar coordinates) and we have the distance.

The fact that we have infinite number of points moving at different velocities just means that we may have to do some integration to get an answer to some questions, such as the distance between two points with different $r$ coordinates. This is the sort of problem that integral calculus was invented for.

 

[Fredrik]

So if a line contains infinitely many points, this implies that it isn't possible to measure the distance between some points on the line, since Planck length is defined as the smallest unit of distance.

The Planck length has no special significance in SR. It may have significance in the real world, but if you're trying to learn SR, you should focus on trying to understand what the theory says, and not let the real world confuse you while you're doing it.