$(1+x)^a$ McLaurin series
For the second question ($x=1$), take for example $a=-1$. The sum is then $$ \left.(1+x)^{-1}\right|_{x=1} = \left.1 + \binom{-1}{1}x+\binom{-1}{2}x^2+\binom{-1}{3}x^3+\dots\right|_{x=1} = 1-1+1-1+\dots $$ which clearly does not converge to $\frac{1}{2}$.
If $x>1$ the series diverges for any noninteger $a$. For positive integer $a$ it actually converges for any $x$ because, as you noticed, there is a finite number of nonzero terms.