Indeed it would. This is something I'd like to see considered; people might well start reviewing more if they knew they were going to get something of a review in exchange. Food for thought, anyway.
Well, it sounds like a good idea to me. We won't know if the scheme will succeed if we don't try it - and if it doesn't work, it can simply be quietly and unobtrusively banished to the void, and its existence forgotten. Besides, I'm sure I could find time to review a random fic each week. Somehow. There must be others who have a spare hour or two somewhere in their week.
Yes, it would, as discussed above. And about the catalogue... the fact that it ceased to be utilised suggests that either awareness of it faded or it got forgotten (I know I forgot about it very quickly). We could always try and resurrect it if we could think of a way to keep awareness going - perhaps by linking it somehow with the 'random fic gets chosen every week' idea? If there is an Oekaki-style splash at the top, perhaps it ought to contain a link to the catalogue, so that if people find that the random fic is not to their taste they're forcibly reminded that a catalogue exists.
However, that causes the problem of perhaps not drawing enough attention to the random fic, so... I don't know where that leaves us. There is also the issue of getting authors to submit summaries and suchlike to a catalogue in the first place; it didn't happen enough before, which left the catalogue somewhat useless (and therefore may have helped it on the way into its decline). Perhaps there needs to be an incentive to submit something? Although you would have thought that increasing your fic's exposure and therefore its readership would be enough of an incentive...
Hm. Plainly, the matter requires more thought. I'm not sure I've succeeded in doing anything with this post other than confusing myself.