any of the centers, you listed, should accept your repair request. and yes they get the parts from manufacturers.
I believe the Mushko was respected in that regard (if I remember older posts on WP about warranty claim or repair request).
And paid repair requests would be initiated sooner than the warranty claim.
BTW if you are not sure about its warranty then read your warranty detail. Breakage might also be covered in warranties if the cost of the laptop includes accidental breakage insurance cost. But this is normally used to be an optional extension when you get a customized laptop. If you purchased the laptop from local market, there would be rare chances that it would be covered in warranty.
(I just heard that now a days these local authorized service centers in Pakistan are not being honored by the manufacturers due to their fraudulent activities of requesting repair charges from customers and in against initiating warranty claims for fake service tags to receive product parts for free of cost)