Irelia, Fervent (Alternate Art)

Yes, if you are Irelia's controller and choose her as the target; countering the original spell or ability does not counter her separate +1 ability. As Irelia's controller, you fulfill her trigger condition by choosing her while playing the original spell or ability.^{412.1412.4355.5355.7400} After the original item finalizes, Irelia's triggered ability is added as a separate chain item.^{383.4.b.2383.4.b.3} Countering the original item afterward clears only that item, not Irelia's separate ability.^426.1.a

Blade Dancer's ability can trigger Irelia once by readying an exhausted Irelia controlled by the same player, but it does not trigger her by choosing her. "It" in "ready it" automatically identifies the friendly unit from Blade Dancer's trigger condition instead of making another choice, so that unit is not a target of the ability.^{355.10.d355.10.d.1} Irelia's controller fulfills her trigger condition when Blade Dancer's ability changes her from exhausted to ready.^{412.1412.4416.1} An already-ready Irelia cannot be readied again, so her ability does not trigger.^{416.1.b416.1.c}

Yes, once, when her controller readies her from exhausted to ready during their Awaken Phase. During their Awaken Phase, Irelia's controller readies every game object they control that can be readied, and changing an exhausted Irelia to ready creates her trigger.^{315.1.b412.1412.4416.1} An already-ready Irelia cannot be readied again, so no readying event occurs and her ability does not trigger.^{416.1.b416.1.c}

Yes, if Irelia's controller chooses her both times; each choice is a separate targeting event, so Irelia triggers twice. Paying [Repeat] makes the spell execute its instructions an additional time, with choices made separately for each execution.^{820.1.d820.2820.2.a} Irelia's controller creates those two targeting events by choosing her once for each execution.^{412.1412.4355.7} After the spell finalizes, one instance of Irelia's triggered ability is added as a pending item for each targeting event.^{383.4.b.2383.4.b.3}