Can I simplify the recurrence T(n) = 2T((n+1)/2) + c by ignoring the "+1" part?
This probably does not change the asymptotic order. 

From $T'(n) = 2 T'(n/2) + c$, you can obtain $T'(n) = \Theta(n)$.

Then you can try to prove that your original $T(n) = \Theta(n)$ by mathematical induction.