Maclaurin Series questions are less of hard and more of tedious. Practice and you'll get better. :)

I have double checked the differentiation, it should be be fine. You might want to check the values of y , dy/dx etc.

As for why the approximation is poor, it's usually the same reason: the x is too big. Approximations get better as x gets smaller.

Lastly, equation of tangent at x = 0 is y = 0.5x because y = 0 and dy/dx = 1/2 when x = 0.