If the target creature you don’t control becomes an illegal target of the first triggered ability before it resolves, Foe-Razer Regent won’t fight and its last ability won’t trigger.

You choose the target of the first triggered ability as it goes on the stack, but you choose whether or not to fight as that ability resolves.

If a creature you control fights during an end step, the delayed triggered ability that puts +1/+1 counters on that creature won’t trigger until the beginning of the following turn’s end step.