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Villains do not make it into your deck for any reason when playing crisis mode.
In most card games, when something says "can't" no amount of "can" will trump it.
Correction: You cannot have villains in your deck unless it says you can pick it up from the destroyed deck or a Crisis card says so. There are a number of cards (such as Signature Trenchcoat and Black Lantern Power Ring) that allow you to grab villains from the destroyed pile. You can only gain villains from the Line-Up "normally" when you have Crisis Black Canary or the Crisis card from Crisis Expansion 2 that changes the rule temporarily. There may be some more exceptions I am missing though.
In terms of Aquaman's Trident, if you gain from destroyed pile with the specific exceptions from the Line-Up, you can put it on top of your deck instead of your discard pile. If you gain Solomon Grundy in those same fashions, you can use his ability to put it on top as well.
The one thing to remember is that you do not gain villains from the Line-Up except when cards explicitly state you take the action instead of destroying the villain per the Crisis rules.