If I use Vilya and I reveal Heed the Dream, can I use the 2nd part of this last card also for free? (the "search for any card if you pay 3 leadership resources")

Vilya:

Heed the Dream:

Because of the word "Then" I think not. You could pay the resources and do it though.

You still have to pay. Here is a thread where similar questions have been answered by the developers.

Notably,

"5. Does Vilya ignore costs on cards that have a built-in cost in the form "cost to effect" phrase like Quick Strike? If so, how does Quick Strike work if you have not exhausted a character?"

"5. No. Vilya allows you to ignore resource match and printed cost with its ability (and anything that is described as “an additional cost”), but it does not allow you to trigger effects without paying their ‘cost’ because these ‘costs’ are not what Vilya refers to. The term ‘cost’ as used in the expression “cost to effect” is not a formal cost, like the kind that Vilya can ignore; they are an informal ‘cost.’ We only call them costs to help players understand that you must do whatever that thing is in order to receive the benefit of the effect. You can think of the difference between ‘costs’ here like big “C” costs and little ‘c’ costs, similar to big ‘A’ Actions and little ‘a’ actions in the game."