Haymaker window and type

To be clear, this was answered on Discord, but posting here:

Haymaker is played during "Before Strikes." This step was codified by BCP. Haymaker's older wording template put it in this phase, as "after" always means immediately after, which is now the "Before Strikes" step.

Haymaker is a strike card as it provides and sets your strike (to hand strike +1 damage). Because it is a strike card is can be canceled with Anticipation. See also this LSJ ruling: usenet.krcg.org/t/qSIV67_nsz8/#m4

To add to kschaefer point here.

Window
"after" and "before" windows in VTES always designate a precise moment, they don't allow a whole period. "before block resolution", means just before block resolution, not anytime before. Same here for Haymaker: after range is determined is that exact moment, not any time after.

Nature
A card that locks in a strike is a strike card. For example, Aid From Bats is a strike card, even if played during maneuvers. Cards that provide a possible strike but leave the choice open (Zip Gun, Hunger of Marduk) are not strike cards.

So indeed,

Can Haymaker be played by a blocking minion after the acting minion has chosen their strike?

No

Is Haymaker itself a strike card and therefore cancellable by Anticipation?

I would have said yes, although I suppose a confirmation there would help

what prevents it to be played in phase 4 was the fact that haymaker affect both initial strikes, to me is clear that this card is a "before strikes" card like immortal grapple