Your satsuma mandarin tree has a couple of common problems. The twisted and distorted leaves with silvery trails are being attacked by citrus leafminer. This is a tiny/near microscopic moth that lays eggs on new leaves. The eggs hatch into tiny larvae that feed on the outer epidermal cells of the leaves, causing them to distort as they grow. This is a nuisance problem that does not impact health of a 6-year old tree, so we don't recommend attempting to spray with an insecticide as frequently as would be needed to control it. 
Photo #1 shows fruit that has been eaten by some sort of rodent--which has removed a circular area of peel that is now infested by sap beetles. Photo #3 shows a fruit on the upper part of the canopy that has been sunburned, because of its position and daylong reception of sunlight. This fruit also shows scratches and peck marks from mockingbirds. You may want to consider tree netting that will keep birds away from the fruit, add a slight amount of shade for summer sun and keep rats, squirrels and other rodents away.

Citrus fruit ripen (get sweeter, less acid) on the tree, not off the tree. Color may progress from green to orange after being picked from the tree, depending on the storage conditions. Peel color and the internal quality do not always match perfectly. You can have partially green citrus fruit that are fine to eat. Citrus fruit store well on the tree; they do not have to be picked after ripe, but may need to be picked early if a severe freeze in November or December is expected.