Bunch of questions;

[fadillzzz]

Firstly, how do you edit the sidebar links?

inc/functions.php Line 256 - 264

To remove, say, adopt an abandoned pet?

You want to remove the adopt abandoned pet feature? just delete the file and any links to the file. I think that should do the trick.

I can't find out how to edit the cash a user has

Manually edit through phpmyadmin (That's what I do)

where I can change the currency name.

I think you have to edit every files that output the text for the currency name with your own

Also, could I disable abandoned pets? :D

I think I already answer that

Was also wondering - can promo codes retire? I'd like to have my pets available for adoption for a month; after that the pet will be retired.

I don't think such feature exist...yet! But you can limit how many times the adoptables can be adopted. There is also a feature to only allow the adoptables to be adopted on a certain date

There's still a chance though for that pet to be brought back as a "rare" for specific people.

Change the promo code, and give them to those "specific" people.

Can a pet be given to a specific user?

Use the Gift System (Not included by default, try to search the forum)

I will be having a special pet just for that one person. Would you recommend sticking to the promo code for this case?

Yes, absolutely! Just limit how many times the adoptables can be adopted

And finally, requiring unique email when signing up

Will require coding skills, but can be possibly done. If I'm not mistaken, the next release will have this feature...CMIIW