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Imagine I have two Alpha Legion characters. I make one of them my Warlord and he gets this trait:

I AM ALPHARIUS wrote:In addition to this Warlord Trait, this Warlord has one randomly generated Chaos Space Marine Warlord Trait from Codex: Chaos Space Marines. If this Warlord is destroyed, you can
immediately select another ALPHA LEGION CHARACTER model from your army to take their place and generate a Warlord Trait for them (including this one). If the mission you are playing grants victory points for destroying the enemy Warlord, your opponent will only achieve that objective if all of the ALPHA LEGION CHARACTER models from your army have been destroyed.

I then use a stratagem to give the second character a Warlord trait:

WE ARE ALPHARIUS wrote:Use this Stratagem before the battle, after nominating your Warlord. Select one ALPHA LEGION CHARACTER model from your army that is not your Warlord and determine one Warlord Trait for it; it is regarded as your Warlord for the purposes of that Warlord Trait. Each Warlord Trait in your army must be unique (if randomly generated, re-roll duplicate results). You can only use this Stratagem once per battle.

I take "Clandestine" from the Alpha Legion tree. When my Warlord dies, I select character 2 according to "I Am Alpharius" and give him a Warlord trait.

Questions:
1. Does character 2 have two active warlord traits at that point?
2. Can character 2 generate a trait from any available tree (eg. generic, CSM, and AL) or is ti restricted to CSM as it was with character 1?

1) Correct.
2) RaW you can generate any Warlord Trait when I am Alpharus transfers over because it doesn't state "randomly generated Chaos Space Marine Warlord Trait from Codex: Chaos Space Marines", which means you can pick any trait the 2nd Character has access to, it doesn't need to be random.