That should not be a difficult fix. Bear with me because I come from the B5 chassis but that looks like one of the two smaller bolts that holds the subframe to the unit. The largest of those three bolts is the weight bearing bolt, the other two are not (this is the one you broke). IIRC the torque value for those bolts is under 50 ft/lb, and the head is sheared.
It is safe to drive.
Get it the hell away from that shop and do the following (either DIY or an indie shop). Mark the center of the bolt with a center punch and drill what is left of the bolt all of the way through starting with the smallest bit that is reasonable and working your way up to the largest that doesnt encroach on the threads. Once it is drilled all of the way through then use a high quality extractor (don't cheap out here, quality extractor bits are crucial). It should be no problem to get that out, I've done it many, many times. Once out, use a jack to support the subframe (very important, its heavy afterall), replace the aluminum piece and the bolts, torque to spec, and you're done.
Do NOT accept anything less than full reimbursement for this repair. The shop knew your car was lowered and chose to accept the work, they are liable for the damage. I would not let them touch the car to try to fix it, just take it to a reputable indie shop and pay them to remove and replace it the proper way. Just don't let them try to weasel out of the bill, they know they are at fault and just like every other shop in the world they will try to blame anyone but themselves, but it is definitely their responsibility. If they still won't pay threaten to file a civil complaint, call the BBB, and ultimately go to court if needed.
Good Luck!
EDIT: I just looked at the third pic, that looks like the trans mount, damn that's a big piece of aluminum. I would probably file an insurance claim, because that's going to be an expensive part. If you aren't comfortable doing DIY's, take it to a shop for the fix, you'll need a transmission jack to support it while that piece is replaced (I thought it was just a plate). I imagine the insurance company will go after the shop for this. Document everything in writing, including follow up phone calls, etc. Just restate everything that you talk about and fax it or mail it to the shop (or both), this will be your evidence. Restate what the shop told you and send them a copy.
And next time, change your own damn oil and you won't have this problem!!
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