Can I use other brand laptop chargers instead of my macbook pro adapter? I recently watched a video on Youtube which claims you may charge your Mac through an adjustable power supply, by setting its output voltage on 18.5 and connecting your MacBook adapter’s wire (the one connected to the MagSafe connector) to the power supply. The circuitry at the top of the MagSafe connector determines how much power your mac is going to draw from the power supply, so even if the power supply supplies more voltage and/or amperage, your mac would not be allowed to draw more than the required power from the power supply. 

Now, I just want to know your ideas on this video and if you consider it safe? Does higher voltage, say, 20 v. hurt the chip located at the MagSafe connector or my MacBook’s logic board? And based on all this, may I use a different adaptor instead of my original mackbook MagSafe charger? An adapter like the following HP charger:

HP PPP016H 18.5V 6.5A Laptop Charger

The link to the YouTube video:
https://www.youtube.com/watch?v=B3w8nwyrdYw
Let's start off by addressing the video you linked to....
It's a video by Louis Rossman of The Rossman Group the owner of a laptop repair facility that specializes in component level repair (repairing logic boards).  In it, he explains why he uses a bench top power supply to deliver exactly 18.5V; so he can monitor the amperage draw.  Suffice to say, it's not a &quot;wall wart&quot; he is using.
The MagSafe Adapter.
The MagSafe power adapter can supply voltages from 14.5V to 20VDC.  However, it's not the &quot;brick&quot; that determines if it can charge or not, it's the actual MagSafe adapter that actually determines whether or not it will interface with your Mac.  The brick just supplies the power.
The connector end is what communicates with your laptop and tells the brick how much power to supply.  Basically, the way it works is that it that it identifies the power supply as having so much capacity (wattage).  If it's enough, it will charge/power the laptop.  If not, it won't accept the voltage.
This is why you could use a 90W adapter on a Mac that came with a 40W but couldn't use a 40W adapter on a Mac that needed a 90W.  This is true with all other power supplies except Apple added some logic into their connector.
Is it safe to Use a 3rd party power supply?
If you know what you're doing, then yes, it is.
Will the 18.5V, 6.5A charger work?
It should because it's supplying 120 watts of power (18.5V * 6.5A = 120W; more than enough to satisfy any Mac.
TL;DR
Yes, based on the info you supplied that power adapter will work.  Whether it's safe or not depends on your skills with DC power supplies and electrical circuits.