Can I take the hard drive out of my bulky PS3 and place it into a slim PS3?

Yes and no; the had drive type is the same, the hard drive will physically work in the new device. However, PS3 hard drives are encrypted, so if you just move the drive into a new PS3 the drive won't actually work; you'll have to reformat it.

Instead use the Data Transfer Utility to transer as much as possible to the new PS3, though some things don't move over; notably PS1/PS2 saves, you'll have to manually copy those to external media. If you can get the old PS3 to work by any means, do this instead if you want to keep that data. Do NOT reformat the drive until you have the data you need (or you think you'll never get it again.)

I am not sure if you have your old PS3 fixed if it will still read the same data; you'll want to ask Sony Support because it depends on the type of repair; if the device is irrecoverable, you can't get the encrypted data back.

Speaking from experience, no.

Last year, my 4-year-old first gen PS3 YLODed unexpectedly (but, frankly, when is it ever expected?) and, being financially flush at the time I just went out and bought a brand-new PS3 Slim.

I poured over various websites and forums of greater and lesser repute and discovered that yes, you could swap hard drives IF the two operating systems were the EXACT SAME version. The game shop manager, whom I (mostly) trust also seemed to indicate this was the case.

So I booted up the new PS3, updated the system files and proceed to willingly violate my warranty by swapping hard drives.

The PS3 did NOT like that. It wouldn't even boot.

So there it was. Four years of music, movies and games - gone.

Well, not completely.

I did refurbish the old PS3 with a $20 heat gun and some arctic silver and got another couple of weeks out of it before it died for good. Perhaps there was some way to transfer the game saves and preferences in that window. I guess I'll never know.