Hi Michael,
first up, you are going to have to take a reality check on your budget versus your expectations.
you mention a budget not exceeding $800 (is that US dollars?) but want "without any kind of errors", "run old gen, new gen without any kind of lag", "is cheap", "run games without any issues".
you are describing the Holy Grail of computing - and it's just unachievable.
to meet your budget, which is already unrealistically tight, and have a good percentage of that budget wasted on delivery charges, you are going to have to consider a 'budget' gaming PC and then you get into the territory of what games it will handle at an acceptable FPS rate.