Illusory Dragon - As it is an illusion, if a creature has True Sight or immune to illusions, does it ignore the fear and damage completely or only gain advantage on saving throws as a creature who succeeds on an Investigation check as per the spell?
The illusion is tangible because of the shadow stuff used to create it, but attacks miss it automatically, it succeeds on all saving throws, and it is immune to all damage and conditions. A creature that uses an action to examine the dragon can determine that it is an illusion by succeeding on an Intelligence (Investigation) check against your spell save DC. If a creature discerns the illusion for what it is, the creature can see through it and has advantage on saving throws against its breath.
Since it uses "shadow stuff" to create the illusion, I would treat a victim with True Sight as having successfully made the investigation check and having advantage on the save vs Breath.
Sickening Radiance - The wording of the spell indicates that the creature only needs to make a saving throw when it enters the spells area the first time or starts in the area to avoid damage, exhaustion, and negation of invisibility benefits. Does that indicate that the creature can travel through the spells area without needing to make more saving throws to avoid taking any further damage or levels of exhaustion. Is that correct?
Yes, if they made the first ST when entering the area or starting their turn there, and they have enough movement to make it outside the area of the spell by the end of the their turn.
Maddening Darkness - It states that "light created by spells of 8th level or lower can't illuminate the area." But the Sunburst spell states that "This spell dispels any darkness in the area that was created by a spell." Which takes precedence.
I think [MENTION=60210]jaelis[/MENTION] has the right of it here. Since Sunburst is instantaneous, it can dispel the Madding Darkness without illuminating the area. This is a nice compromise between two equal level spells as targets within the Madding Darkness area will not be subject to the Sunburst effects, but the Madding Darkness will be dispelled.