If I do not use any cold skill (crusader), but a weapon with cold damage, will the Chill effect of Iceblink be applied to every attack in order to reliably trigger the damage enhancement of Bane of the Trapped?
You need a cold skill to proc the Ice Blink effect. Weapon damage will not help.
No, in general it will not. The cold damage from your weapon is converted to the damage types of whatever runes you are using for your skills. The only way to proc it outside of using cold skills would be through a basic attack.