No teleportation happens. Sokarul, I am sick of telling you this. You insist on bastardisation because you can't cope with a working model. Nothing special happens at the equator, you know this, you have been told it multiple times. Why are you so scared you must resort to obvious lies?
"If all matter is stationary, how can various points recede compared to each other?
If aether flow doesn't have effect on mass, than how can it have effect on mass, creating the feeling of not falling down, holding things together, etc. ? Everything you can observe in real life."
Stationary with respect to the point they occupy in space. You can be stationary with respect to a train, but not with respect to what's outside. Reference frames are needed to define stillness or motion. What we observe in reality is mass interacting with mass. The Earth below us is caught between two flows, so a stationary object with respect to aether will move towards it, for example.
Flow is a good analogy. Take a waterfall: taking a reference frame in which a particle is stationary would have the Earth rushing up towards it, which is what happens from the perspective of the aether. From the perspective of the ground, which we are in, the particle is falling towards it.
The easiest way to think of it is to picture the aether as free flowing water, where each molecule is a coordinate point in space. A stationary object would occupy a fixed coordinate point, so it follows the motion of said coordinate point. The object itself, however, does not move. When the water flow hits another water flow, the object might strike another: this is mass interacting with mass.
"If all matter would come towards the Earth, within a certain distance, how come we don't observe it all the time? Like falling stars all the time, larger objects, etc.?"
You'd have to get to the star part of the model to see the details, suffice to say they're in what are known as aetheric whirlpools: natural formations where the dominant motion is parallel to the Earth's surface. There is an upwards flow to fill the low concentration left in the wake of the downwards flow which maintains this (again, the details are given in the relevant section: you've only read the definition of aether, the applications take more work).
"Anyway, in a 3D aether, why did dust concentrate to two disks, rather than a globe? If aether would really flow to this low concentration point, which is in the centre of Earth, in a 3D space/aether it would collect all dust from every side, which would create a sphere. Nothing explains the creation of flat disks in a flowing aether from the definition of aether (1. point of the theory).
Also, are the other observable objects in aether (stars, galaxies, planets) all have low concentration points of aether, or only Earth does? If they do, why don't we observe flat disks in space, but spheres?"
I'd recommend continuing reading. The former's answered in the FAQ, the latter requires you to understand the make-up of stars and planets in the DE model. If you are ok with the definition of aether now, then the rest should follow easily.