Verify Points on curve secp256k1
Your computations are correct.

For each valid $x$ coordinate there are 2 valid $y$ coordinate, $+y$ and $-y$, as you correctly said.

But note that numbers in a field $\mathbb{F}_p$ are usually written $\mod p$, which means they lie in the interval $[0,p-1]$, so without the "-" sign.

So, when you write the $y$ coordinate of your $-10G$, which you call $10*G (Inverse)$ but obviously $10*(-G)=-10*G$ so it's equivalent, you write it as:
-62070622898698443831883535403436258712770888294397026493185421712108624767191
but this can be written as a field element as p - 62070622898698443831883535403436258712770888294397026493185421712108624767191 with $p$ being the prime defining the field.
This results into 53721466338617751591687449605251649140499096371243537546272162295800209904472 .

Plugging the coordinates into the curve's equation is the correct way to verify that points are on the curve, and yours are on the curve.

For example: your $2G-10G$ and $8G$ share the same $x$ coordinate, in fact, $2G-10G=-8G$ so they are opposite points. You can get the value of the $y$ if you subtract one's value from $p$. E.g. 

p - 41749993296225487051377864631615517161996906063147759678534462689479575333124 
= 74042095941090708372193120377072390691273078602492804360923121318429259338539