Specifically, if I summon "Trishula, Zero Dragon of the Ice Barrier" and non-targeting banish a Blue Eyes spirit dragon and two other cards, can the Spirit Dragon chain 2 tribute itself, summon Azure Eyes, and the azure Eyes will avoid banishment?
Azure-Eyes Silver Dragon can be banished in this situation
As you noted, Trishula, Zero Dragon of the Ice Barrier has an effect which banishes cards without targeting them. However, this is not only a way to work around effects which prevent targeting; you don't need to choose or declare which cards are being banished until the effect resolves, and there is no opportunity for your opponent to activate anything once you've made your choice (there are other categories of effects which could apply, but none on the listed cards). Thus, there are two options for your opponent:
Your opponent does not chain Spirit Dragon's effect to Trishula's effect. In this case, Trishula's effect resolves you can banish Spirit Dragon and up to two other cards your opponent controls, and your opponent does not get an extra chance to activate Spirit Dragon's effect after you have made your choice.
Your opponent tributes Spirit Dragon to chain its effect to Trishula's. In this case the chain will start resolving, your opponent will summon Azure-Eyes Silver Dragon with Spirit Dragon's effect, and then Trishula's effect will resolve and you can banish Azure-Eyes and up to two other cards your opponent controls. As a side note, if Azure-Eyes is summoned this way and then banished its summon effect will not activate since it will not be on the field to start a new chain after Trishula's effect resolves.
In either case you could also choose completely different cards to banish, including fewer than three or none at all.