Thanks for the pictures, they do help a bit. I think it is a distinct possibility that you actually have a Chinese Elm. They are often confused with Zelkova, sometimes deliberately so. Compare the two side by side, the shoot on the left is a Zelkova, on the right is a Chinese Elm.
I've noticed that the mold has decreased slightly, there is none on the trunk anymore although there is a white residue left behind.--The soil has remained damp, even on the surface, since it's initial immersion, so I haven't watered the soil. Should I still spray the leaves?
Resist the urge to water until the "dampness has lessened" I have never bothered to mist my Chinese Elms, in fact this may cause problems by promoting "Black Spot" fungus, particularly on new growth.
A part in my book says that signs that trees need repotting include "roots that appear like a coconut-fibre mat", which I think is apparent in the left hand side of the first two photos (right hand side in the third)?
I think that the author may be attempting to describe a root-bound situation. The fibers that you pointed out are not roots.
I've had it indoors in an attic conversion room, on top of a bookcase. It's a well ventilated room with dual window aspect. However, after reading through the stickies and my book I realise that even though the label says "indoor plant", I should probably keep it outdoors.
I keep all of my trees outside with the exception of tender plants that come inside for the winter. If this is a Chinese Elm (sub-tropical) some growers do keep them inside over the winter. I prefer to allow mine to go dormant over the winter. If you are able to find a spot outside that gets some sun and some shade. The shade of larger tree would be ideal. Gradually, after you ensure that the tree has survived its recent trauma, it can be move to a brighter location.
The soil quality is currently slightly gritty on top, but under that it's like normal soil. I've got some Bonsai Compost to use for repotting if you think this would be okay?
I don't like the looks of the soil at all. Most deciduous trees are best re-potted in the spring. If it is a Zelkova this is the way I would go. If it is a Chinese Elm there is another option. As I mentioned some growers keep them inside for the winter, if you intend to follow this method it is possible to allow the tree to go dormant in the fall and then re-pot before bringing it inside for its "false spring".
I hesitate to suggest that you take this approach because we have not definitively identified it and because of your inexperience with bonsai. On the other hand you will need to re-pot it eventually, next spring at the latest. Can you post a picture of the bonsai soil that you have on hand. Be aware that even if it is suitable it will most likely need to be screened to remove the fines. I use a 1/8" screen to remove the smaller particles.