I can see why they made into it 4 individual spaces, rather than one giant space like later mansions.
I've only toured the Garage, but have yet to see the other 3 Mansion spaces.
I showed a user, who was giving tours of his spaces, the Batcave, which many don't seem to have.
The clothing rewards in GZ Burger, can also be acquired by going to the Granzella store in the Mall. Both stores have a free section.
@Duke if you need a tour let me know which spaces you would like to see
Other than seeing the rest of the Mansion, no idea. The problem with tours is they're time consuming.
As for the Batcave, I do give tours of it since you need a Batman: Arkham Asylum save to acquire it otherwise. I left the Tardis in there since I took a screenshot days ago, but I've been in the Tardis when someone used it for a clubhouse. Should be able to enter the Tardis if you own the personal space.
I could also show Casino VIP Suite (Not Penthouse, don't have it yet), Ansada Fone (the one from Trap-O-Matic).
I think I forgot to mention I have added the Aurora Showcase Apartment as well as Club Luminosity to my collection of spaces. On a side note I picked up the personal dining experience tables that offer rewards as well for those interested. 8)