Lorentz contraction and magnetism I’m trying to learn about the Lorentz contraction and it’s relation to magnetism. I have two questions about them.
1-  I watched the “veritasium” video about the relativistic explanation of the magnetism(https://www.youtube.com/watch?v=1TKSfAkWWN0). 
I can’t say anything about the reliability of the “veritasium channel” but I know it’s very popular. Here is one thing that puzzled me in the video:
At 1:28, the electrons of the wire is moving relative to the positively charged cat. So, from the cat’s perspective, the moving electrons should be subject to lorentz contraction and there should be a net force on the cat. But in the video, there isn’t.
At 1:50 the cat starts moving with the electrons. This time however, again from the cat’s perspective, the positive charges appear to be moving and because of the lorentz contraction, there is a net force (magnetic force) on the cat. 
This seems to be same as the explanation from Purcell described here (Magnetism as a consequence of length contraction): http://physics.weber.edu/schroeder/mrr/mrrtalk.html 
But how does this make sense? When the “test charge” is stationary in the lab frame, the moving charges on the opposite should have been contracted but they are appearantly not.
2-  Do all the contraction/dilation effects of the special relativity work in tangential directions?
If a charge q1 is moving fast directly towards a stationary q2, there would be no magnetic force acting on q2. No contraction of q1 on the q1-q2 axis.
If the charges q1 and q2 are moving together on the same axis, again there would be no magnetic force acting on them. No contraction on the q1-q2 axis.
And from the form of the Biot-Savart equation we can see that the “length contraction effect” is dependent on the sinus of the angle between the directions of the moving charges.
These imply to me that the “length contraction” of an object A, could only be due to it’s tangential velocity component relative the point B (the circular motion of object A, around the center B). Is this right?
 A: First thing you need to understand is that you do not understand length contraction. So here is a short lecture about that topic:
Let us consider a 1 kg mass being accelerated by applying 1 Newtons force on the other end of a 1 m long rope attached to the mass. 
The frame from where we observe is the inertial frame that is instantaneously co-moving with that end of the rope where the pulling force is applied. 
The instantaneous acceleration of the rope-end is 1 m/s^2, as common sense suggests. The instantaneous acceleration of the mass is slightly larger than 1 m/s^2.
In our frame the instantaneous velocity of the rope-end is zero. The instantaneous velocity of the mass is non-zero. In other words the mass is moving towards the rope-end in our frame, in other words the rope is contracting.
And now let us consider two 1 kg masses being accelerated by applying two 1 Newtons forces on each of them. The accelerations of the masses are the same, the velocities of the masses are the same. Objects that have the same velocity do not get closer to each other or farther from each other, in other words distance between the objects is not changing. In other words no length contraction is occurring. 
A: 
I can’t say anything about the reliability of the “veritasium channel” but I know it’s very popular. 

It's popular because it's usually right.

At 1:28, the electrons of the wire is moving relative to the positively charged cat. So, from the cat’s perspective, the moving electrons should be subject to lorentz contraction and there should be a net force on the cat. But in the video, there isn’t.

If the wire is part of a circuit, the "supply" and "return" wires are actually at different potentials, and there's an electric field in the gap between the wires which would, in fact, push on any free charges. Here's a diagram from Wikipedia that shows the electric field $\color{red}{\vec E}$, the magnetic field $\color{green}{\vec H}$, and the Poynting vector $\color{blue}{\vec S} \propto \color{red}{\vec E} \times \color{green}{\vec H}$ which shows the direction of power flow through space.  For real direct-current circuits, the electric field in the empty space around the transmission line is small and it's usually safe to ignore.  Figuring out how small is a useful homework problem.

By Chetvorno - Own work, CC0, Link

Your observation that charges approaching each other head-on won't be able to detect this Lorentz contraction, and won't feel a magnetic force, and that this is consistent with the Biot-Savart law --- that's all correct.
A: Alternatively consider a rotating wave of an electron. It rotates about an axis in X direction. The rotating electric wave portion creates a rotating magnetic portion and vice verse. Rotating fields generate the charge. Like photons the rotating wave electric field must be perpendicular to the magnetic field and both must be perpendicular to their motion. A rotating wave in a stationary spot will generate a spherical electric field and bipolar magnetic field. Electrons move along their axis of spin, so when put in motion the electric and magnetic field must incline to remain perpendicular to the helical motion of the wave. The inclined fields effectively shorten the length of the electron. That’s length contraction. Just as important, the magnet lines incline and induce a right handed magnetic field.
Of course the cyclical change in wave function leads to quantum mechanics. In the least it is another way of looking at the universe. It clearly shows the direct connection between length contraction and magnetism.
