1. Mid-afternoon for the Aye-aye and early morning from the Potto, although you can see both all day long, and especially the Aye-ayes are almost a guaranteed sight with a little bit of patience (in recent visits, the longest that I have had to wait at the enclosure to see an Aye-aye is around ten minutes, although it is of course possible that I have just been getting lucky).
2. Cayenne Caecilians are very much on-display, in a tank to the left side of SLoRA opposite the giant salamanders. They share this tank with Rio Cauca Caecilians and Bristlenose Plecos, and due to the number of individuals held and how active some of the larger individuals are, you are guaranteed to see them. Regrettably, as mentioned by
@cerperal , the Congo Caecilians are much more of a challenge, with their main enclosure (in a glass window next to the old Aquarium) now being boarded up. As you enter SLoRA, there are enclosures for Laos Warty Newt and Big-headed Turtle to your right; if you look behind these, some offshow tanks are visible, with Congo Caecilians being kept in a few of them. Sadly, these are very challenging to see.
3. Yes, they are still in the same exhibit by the lions, but are indeed very shy. Sadly, I am yet to see one myself.
4. Not sure about the goby, but the squirrelfish and damselfish were definitely kept in Tiny Giants on recent visits. Like the rest of the fish in said building, they can be found in the large coral tank on the lowest level of it. Unfortunately, as cerperal mentions, it can be very busy.
5. They are shy, but love grazing on the open grass, so it is unlikely that you will miss them. The only risk is that they can be startled by large crowds, so visiting either in the early morning or late afternoon will increase your chances slightly, as there are fewer visitors, but personally I have never had trouble seeing them.
6. Most of the other rarities have already been touched by cerperal, but to be more specific about the other rarities in Blackburn Pavilion, they include the only Scarlet-chested Sunbirds in captivity, the only Splendid Sunbirds in Europe, two pairs of Collared Trogon, Ultramarine Grosbeak and Purple-naped Lory, with all except the lories being in walkthrough spaces. He is also correct in stating that almost every species in SLoRA is worth seeing, but in particular don't miss out on Europe's only Lake Oku Clawed Frogs and Round Island / Telfair's Skinks, of which London is one of just two European holders.
Enjoy your visit and feel free to ask any further questions!