To start with you are quoting a version of Xeno's paradoxes
The conclusion to these is that the statement
If you throw a rock to a tree, the rock will go half the way, then half the way, then half the way and it will keep going the half of the remaining way. Therefore, the rock will never hit the tree
is absurd,, and patently wrong since the rock will hit the tree and its trajectory can be calculated to great accuracy.
I do not see how it could apply to time except to say again that some conclusion is absurd.
Now you start with
I have always thought that the time is discrete (jumping), not continuous and the shortest time is Planck Time
All the data we have of physics of elementary particles and astrophysics are clear that time is a continuous variable, and that is a lot of data. The statement has not been falsified. The shortest delta( time) is 0, called synchronous if it is between two events.
Paradoxes appear when one mixes up frameworks of reference. It is true that there exist mathematics where a series has an infinite number of steps. Any functional form of mathematics can be expanded in similar series. The confusion comes from thinking that because a mathematical function , a parabola in the case of the rock, describes/models a physical phenomenon, the physical phenomenon is created my the mathematical form and should materialize any expansion , series formulation automatically. Physics uses mathematics, mathematics does not create physics as far as we know .
How come the self-appearing virtual particle duos live shorter than Planck Time if it is shortest possible time?
Virtual particles are another such phenomenon. It is unfortunate they are called particles, they are just formulation under a lot of integrals. It is the end result of the calculation that has physical meaning. They are called particles, but they do not have the mass of the homonymous particle, they are off mass shell, a mathematical construct useful in calculations that carries the quantum numbers but not the mass.
So the bound is not the Planck time, a red herring in this case, but the Heisenberg Uncertainty Principle