It looks like the latest version of OS X that you can use on that Macbook is OS X 10.7 "Lion". Your best bet would be to find the disk image somewhere and burn it on a disk or put it on a usb drive and then boot from the disk image and then wipe the drive and start over.
I have an A1181 with 10.9 on it. It depends, they recycled the model number for several machines with different guts and the same outsides. They got different version numbers. Mine's a
Macbook 5,2. Everything from a MacBook 1,1 to a 5,2 was an A1181. Except for the Macbook 5,1 which was the original Unibody. The 6,1 and 7,1 followed, until they were discontinued in favor of the Macbook Air.
Except the A1185, which was the A1181 in black plastic. (Same, otherwise. I keep meaning to buy a 1185 keyboard one of these days and swap the keycaps. It'd look awesome.)
None of them, however, support cloud boot, wifi boot, or internet recovery.
As for 10.8 or 10.9 compatibility, the easiest way to tell is the video card. If it's a Geforce 9400M, yes. Intel GMA950 or X3100, no.
Anyway, OP:
Find another mac. Log into App Store, buy Lion (Or Mavericks, depending), download, use SuperDuper to clone to a thumb drive. Insert thumb drive, turn on old laptop. Hold down option/alt key, boot from thumb drive. Select HDD, reformat, reinstall.
Is your HDD working? If not, it's a standard 2.5" SATA HDD. Replacement directions are on iFixit.
Cousins, hot or not, frequently have hot friends who are not your cousins, and know you well enough to be effective wing(wo)men. Get to work, don't diddle your cousin.
All the problems in this thread are now solved. Good night.