it didn't have anything for the 328P

The purpose of an application note is to show you a concept so that you can understand it and take it to your particular application / target.

is that feesable enough to?

It is definite feasible enough for some.

TI didn't seem interested in cross platform support.

The more you target commercial users, the less you care about cross-platform support.

Maybe it's just cheap Chinese junk...

Or it is being mis-used.

But as these sources evolve,

Life isn't supposed to be perfect.

I was trying to study the relation between the applied voltage and crystal skew.

What voltage?

But it seems the board has resonator which is not very accurate.

A brain-dead decision.

How can I solve this problem.

You can replace it with a crystal oscillator or an external clock.

My aim was to obtain less than 5ppm error with the arduino

Very easy to do.

without using any external clock.

That can be challenging. Crystals guarantted to be in that range are expensive and not very common. A far simpler / cheaper solution is to use external clocks. TCXOs for example are that much more expensive than a crystal.

As far as my schematic goes, is this correct?

Something like that will never work: there is no signal loop.

You need to common the two circuits.

Before you embark on buying new devices, get the circuit right.

Eventually, everyone will vote conservative, once they're no longer ignorant.

One way to defeat an unsustainable scheme is to expand it as fast as possible, so it collapses quickly.

It certainly can be made to work.

Some suggestions:

1) With high current switching, you want to use a switching mode controller, rather than a linear controller.
2) You want to check the device's thermal resistance and make sure that you aren't off-dissipating. For soic/no thermal pad, you should try to stay within 1/4w, or 1/2w max (indoor, open air flow). Even with thermal pads, I would be extremely concerned pushing the device to 1w of dissipation - those things have thermal resistance of at least 100c/w. So at ambient of 50c, you are at the limit.
3) You should think hard about if you want to switch at 20khz.
4) the ringing is due to reactance. One way to solve it is with slew-limited drivers; or to a bead on the output line.

I would look for those dc/dc converters (SimpleSwitcher from National for example) as an alternative here.

There seemed to be a faulty cable, I just reconnected it and it works now

How dare you? Providing an example where the fault is not decoupling caps?

shut off OE, update the shift registers and switch to the next row, then turn OE back on.

You can get flickering with this approach, if a lot of data is transmitted.

A better approach is to use the latch.

What's good is highly subjective.

They can be easily compiled to the Arduino, but I was rather looking for a 'library' where you could you know..

How hard would it be for you to stripe out the "shell"?

it was even "on the level"...

What he was doing is totally unnecessary, unimpressive, overly complicated and fairly inaccurate.

With NO external hardware (including a wire), you can achieve better timing than he did.

At 16Mhz, it takes 1us to execute 16 nop instructions.