5V 8A wall adapter -> arduino?

I'm using this 40W adapter to power 5m of LEDs, but it seems like I could just split some of that into the 5V pin of the Arduino, correct?

I know the Arduino only wants a little bit of current,so I'm thinking it should only draw what it needs. I don't want to overwhelm it or anything.

Also, pretty technical, but I cut the ferrite inductor thing off the end of my adapter while setting up this whole LED project. I figured it wasn't a big deal. I know this might make the input current a bit more volatile, could that be a problem?

I know the Arduino only wants a little bit of current,so I'm thinking it should only draw what it needs. I don't want to overwhelm it or anything.

Yes, that's correct. Just like if you connect another LED (with the proper current-limiting resistor).... That additional LED would only draw another 10 or 20mA.

Also, pretty technical, but I cut the ferrite inductor thing off the end of my adapter while setting up this whole LED project. I figured it wasn't a big deal.

That shouldn't hurt anything related to the Arduino, and it probably won't hurt anything else either.

That bead is there to reduce radio-frequency emission from the power supply's oscillator. Without it, the power supply might affect your radio or TV reception, etc., under certain conditions. Without the bead the power supply might not meet regulatory requirements. (In the USA, that would be an FCC regulation.)

Often, that's something added as a fix after the product is submitted for testing and it fails the regulatory requirements. Or, some manufacturers may just add it as a precaution.