1.Can I revive a hero when I stand as the same space as the KO hero ?

2.Can I search a search token at the same space ?

In fact , I usually consider that it's legal.

If not , it seems to be a bit of stupid. But this rule effects somewhere sometimes.

3.Can Mistpeak's ability trigger Gold Rush and Lure of Fortune ?

1.Yes but Not in the same space ( adjacent space if you stand in the same space where hero died.)

2.Yes same space or adjacent if you have an action to do so.

3. No for both since they are both require a search action and Mistpeak perfoms an attack action in order to activate his ability.

I disagree on point 3.

I think, Thaiden's Hero ability would indeed trigger Goldrush and Lure of Fortune because Trail of Riches has been confirmed to do so.

Both, Thaiden's Hero ability and Trail of Riches have the identical phrasing "search a search token within 3 spaces of you" and are considered search actions (just without the word "action").