Datasucker tokens last until you pass the ice. They clear at the same time Prey triggers (when you pass means the encounter is over) its conditional ability. Removing tokens happens first, as it is a game state change, so they cannot be used with prey’s conditional ability.
The Prey/Datasucker interaction was also discussed in the Flashpoint thread, based on this post, I believe @Therad got a confirmation from Damon that “passing” happens right after the encounter/approach phase ends:
It would be great to get Damon’s full clarification (I didn’t see it on this Tweeter feed, so maybe it was a DM or a response to FFG’s rules question submission form).
Can you trigger the shuffle with replicator if you have no more copies of the installed hardware in your stack?
The text: Whenever you install a piece of hardware (including Replicator), you may search your stack for another copy of that hardware, reveal it, and add it your grip. Shuffle your stack.
I feel like I remember a similar situation for firing smc and not having the credits to install anything, but I couldn’t find the relevant ruling.
Basically I would like to keep a spare copy of an hw in hand so that I can spy camera and, if I don’t like what I see, trigger a shuffle and try again.
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You can do it because shuffling your stack is a change of game state.
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Passing is by definition the end of the encounter.
Damon: “You cannot currently be encountering and have passed a piece of ice at the same time. It is the transition from Step 3 into 2 (approach next ice) or 4 (approach server) that meets the “pass” condition and that is when Prey resolves, while the strength reduction by Datasucker ceases at the end of 3.2.”
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So Prey + Ice Carver shenanigans ends here too 
Yes, since DS and carver works in the same time span. 
Not quite. They end at the same time, but Ice Carver begins earlier.
Prey does work with Scrubbed!
The first piece of ice encountered each turn has −2 strength for the remainder of the run.
And Null!
Once per turn, when you encounter a piece of ice, you may trash 1
card from your grip. If you do, that ice has -2 strength for the
remainder of this run.
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Security Testing vs. Patron.
Runner has one of each installed and, at start of turn, designates both on, say, archives.
When runner makes a successful run on archives, does he/she get to choose which benefit to take (2 creds or 2 cards)?
Or does the game crash? 
How do two Security Testing on the same server work?
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?
The runner has a Security Testing AND a Patron installed.
At start of turn runner designates Security Testing on archives.
At start of turn Runner also designates Patron on archives.
Runner makes a successful run on archives.
Can the runner choose between the Security Testing benefit (two credits) or the Patron benefit (two cards)?
edit: OH! Were you being pedantic? I don’t know. Can one designate two Security Testings on same server?
You can only do one replacement effect.
I don’t think anyone expects to get both 2 cards and 2 credits for making a successful run on that server. I do think it’s a half-decent valid question whether the replacement effects overrule eachother using timestamps (like they happen to do on jinteki.net) or whether the player controlling the trigger makes a choice at the moment that the replacement effects both raise their hands.
If your first click is going to be Making an Entrance, and Archives is the only vulnerable server, this does make a real difference since you would prefer to delay the choice on whether to take cards or credits until have you’ve seen what Making an Entrance shows you.
I figure we probably have a blankety rule about choosing which replacement effect applies -somewhere-, but off the top of my head I’m only familiar with ones about ordering triggered abilities, which are fundamentally different from replacement effects. 85% sure it’s on the books somewhere, but not sure where. Wherever it is, that should be the thing you pithily point to rather than insisting it’s obvious (well, maybe in addition to insisting it’s obvious, whatever).
Like all abilities that meet their trigger condition at the same time, you get to choose the order in which they trigger.
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What’s the current rules analysis of Imp?
Once per turn, you may remove 1 hosted virus counter to trash a card you access at no cost (even if it cannot normally be trashed).
That’s templated as a constant ability, right? But it specifies an implicit time when you’re allowed to use it by saying “a card you access”, which suggests that you have to be accessing the card in question. But I’m not really sure why that implies that it happens after any “When you access” effects.
One of the first things I learnt about constant abilities is that they take priority over triggered abilities! Under a naïve reading of the rules, I could see assuming that you could access an IAA’d junebug, and use imp before the when-you-access trigger happens.
Imp has been discussed quite a bit. Use the forum search bar to look back. (It’s also in the FAQ.)
No, I mean, I get how it works; as of the FAQ we know that any triggered access abilities happen first, that’s fine. What I’m asking is if we have a rules understanding of why, given the wording? Does this tell us anything about how constant abilities with things-you-could-argue-to-be-timing-restrictions-on-them work in general?
I had a search upthread, and didn’t see anything talking about that question.
It has a condition. So it’s a conditional ability.