Approximation results on scheduling under an uncommon constraint
Even without the additional constraints you have the unrelated machine scheduling problem with precedence constraints is not well understood. If I am not mistaken nothing better than a polynomial factor approximation is known. It is an important open problem in approximation algorithms for scheduling problems. Some positive results are known when the precedence constraints form a tree. See the following paper. http://www.springerlink.com/content/p8321031hk718413/

Can you solve your problem in the simpler case when the machines are identical or related?