Yes. In many cases like this the right answer can be found by looking for "keywords". In this case the keyword is "draw": the Basilisk does not allow you to "draw" additional cards. If an ability does not use the word "draw" then it is allowed even if it involves adding a card to your hand. In this case, "searching" your deck or discard pile and "adding" a ...
From what I understand, it just goes back on the top of your deck and you'll just know at least one of the cards you get on your next turn.
If you don't need to discard a card, would the card sit at the top of the deck or move into your hand for the start of next turn?
The standard interpretation of "reveal"...
There's a comprehensive FAQ compiled from answers to questions from the game designer
Q: If a Villain blocks part of a card effect, can you or must you execute the rest of the action?
A: Yes. For example, Lockhart says “Draw one card, then discard any one card.” If the Basilisk is in play and you choose to play Lockhart, you would not draw the card ...
You will not gain the heart or remove the control.
Andrew Wolf, the game designer from USAopoly, cleared this up over on BGG:
... if [a Villain] has an ability preventing the Heroes from doing something, they do not gain the reward from defeating another Villain.
Yes, you can search for cards when the Basilisk is out.
This is confirmed by game designer Andrew Wolf on BGG (although he is talking about the Charms Proficiency, but the meaning seems clear):
Searching your discard pile for a card to put in your hand is not considered drawing, and as such, not prevented by the Basilisk.
BGG User David Jones used that ...
In this BoardGameGeek thread here another player had the same questions. The only problem is that they are playing the full game, and there are things that do not apply to you (moreover there will be spoilers that maybe you want to avoid).
So I'll try to sum up his rules that apply to Game 4:
To make the game shorter, I would say start on location 2, and ...
The way I interpret the rules is that you can use the zap. Here's what the rulebook says when being stunned:
Discard any or tokens you may have saved on your Player Board
from other Heroes’ turns.If it is your turn, you may still play cards and take actions with whatever you have remaining after being Stunned.
The above was found on page 18 of the ...
From my previous answer regarding the situation, I don't think you can avoid the stun by gaining the heart. As soon as you play an ally the horcrux would trigger, causing the stun to happen and stunning states:
You cannot lose (or gain) [heath] any more this turn.
Then the card effects would happen. Here's a BoardGameGeek thread discussing this exact ...