Yes, it can. White Night Dragon can only negate Spells or Traps that target it. For one, Solemn Judgment does not 'target' anything, and for two, WND's negation effect is only active while it is face-up on the field after being successfully summoned. Solemn Judgment is preventing its summon.
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You can use Solemn Judgment on the summon of one of your own monsters, if you wish.
To Special Summon Judgment Dragon, you must have at least 4 different Lightsworn monsters in your Graveyard. Once this criteria is fullfilled, you can Special Summon it from your Hand onto the field.
No. Monsters can special summon themselves in two ways, there are monsters that special summon themselves without using the chain, such as Cyber Dragon, and Dark Armed Dragon, but also Synchro Summons. Then there are those that are special summoned by the resolution of an activated effect. To negate the former, you wait until the new monster is placed on the field. Summon negation cards like Solemn Judgment, can then be used on it. To stop the summon in this latter case, you have to negate the activated effect - once it resolves, the summon is successful and can't be stopped. DD Scout Plane summons itself on resolution of its Triggered effect. Solemn Judgment cannot negate activated monster effects, so cannot be chained to this activation. Once the effect resolves, DD Scout Plane is considered to be successfully special summoned, so can't be negated. If the monster is being summoned by an activated Trap or Spell card, Solemn Judgment can be chained to these. But it cannot stop a summon from an activated monster effect, which is why Solemn Warning became so heavily used, because it can negate such monster effects.
no, dragon master knight can only be fusion summoned.
No, Divine Wrath can only be used against effects that activate. Most of the time, when a monster has a way to special summon itself, this is an inherent special summon, just like a normal summon, it does not use the chain. Because it doesn't use the chain, you can't chain an effect negation card to it, though you can use summon negation cards like Solemn Judgment. If the special summon is an activated effect (like Dandylion's effect to special summon tokens, or Gorz's effect to summon himself) then these can be stopped using Divine Wrath. But Cyber Dragon's special summon cannot.