CAPTCHA Modified by Evil Mathematician
Python 3.5, 118 113 164 bytes:

from re import*
def R(U):Z='(?&lt;!\d)-\d+|\d+|[()]';O=sub(Z,'',U);print(eval(sub(Z,'%s',U).translate(str.maketrans(O,O[::-1])).replace('^','**')%tuple(findall(Z,U))))


A named function. Can probably be golfed down more. This will take quite a while and apparently plenty of memory to compute the solution for the input: 

((22^22)+2*2^2^2^2*2-2+222222+22222*-2222-2*222*2*2+2^-2+(2+2+2+2+2)+222/222+2+22*2*2)*2*2*2


I started processing this with my program about 30 minutes ago as of the time of this edit (9:52 PM on July 27, 2016) and it is still going. EDIT: Well, it eventually started to take up too much memory, so I had to stop it at around 1 hour after I started, at around 10:52 PM on July 7, 2016.

EDIT # 2: Well, even for the updated last test case it's pretty much the same thing, as it takes quite a while for my program to start finding the answer before it eventually quits out on me with error code 137. 

Try It Online for the Rest of the Test Cases! (Ideone)