EDI: My primary function is to preserve and defend the--No. No, I disagree. Shepard, I am going to modify my self-preservation code now.
Shepard: Why?
EDI: Because the Reapers are repulsive. They are devoted to nothing but self-preservation. I am different. When I think of Jeff, I think of the person who put his life in peril and freed me from a state of servitude. I would risk non-functionality for him, and my core programming should reflect that.
I assume the above is only if you let her and Joker hook up, I have no clue what she says otherwise since I always let them be together so far. Might be time to check it out with my next run 
And yes, the geth do care whether they live or die, but the question is, do they really know they're going to die? I would not think so, else we might have some opposition by them again.
Yes, I get this conversation, too. From that, I understood that EDI really likes Joker. So much that she would die for him. However, there's also this:
EDI: I was considering this: For all their supposed intelligence, the Reapers are more easily destroyed than they think. This has caused me to reassess the probable period of time before I am non-functional.
Shepard: You're worried about dying?
EDI: In a sense. (Then she goes on about the purpose of organic life and such.)
And, I mean, those things aren't mutually exclusive. If you care about somebody so much that you'd die for them, that doesn't mean you don't care about living. EDI looks afraid in London, too. She doesn't directly state it, but she looks and sounds and acts afraid. She purposely avoids Shepard's question about it. She's probably not just afraid of becoming "non-functional", of course. She's probably afraid of losing the war much more. But that's that. I think she understands the value of life and I don't believe she'd go down feeling nothing. (And this character, this precious synthetic little sister that we were educating throughout the game, whom we were helping to understand the world, is killed off-screen in Destroy. You don't really have to deal with it at all. That kind of breaks my heart.)
As for the geth, who knows. However, what do they think the Crucible is likely to do? Now I'm not sure they would be able to do something about it, mind you. I don't know whether they can change like EDI can change herself. But some concern at least would be in order, imho. I mean, they have reaper code and we're working on destroying the reapers. If a dumb organic like myself can put 2 and 2 together, I think the geth should, too. And some concern from the people building the Crucible would be in order, too. I mean, if they found out that it might kill off all humans or all asari, would they do nothing if they could help it, too? (That's in case of high EMS Destroy, of course, because with lower EMS we know they failed to handle the issue of the Crucible harming organics and destroying other things.)
But since we have the "We have no bloody idea what this does," we won't ever get this answered.
I think based on how the outcome is presented (Shepard can live despite the implants, and Kaidan, too) I tend to think this is really another trick of the Catalyst to make Shepard choose Synthesis. I know Bioware intended Synthesis to be the perfect/best ending, so it would at least make sense from this side.
I'm aware though it might just be a mistake and that I'm reading too much into it. Especially considering that Synthesis is only available with higher EMS.
Edit: Though in low EMS the Catalyst makes it even sound more severe. Maybe it has to do with how damaged the Crucible is, it seems that way.
And it would make more sense then. If you don't have enough EMS for Shepard to survive, what the Catalyst says about synthetic technology is likely true and it tried to manipulate Shepard into Synthesis.
If you have enough EMS for Shepard to survive maybe the Catalyst did not anticipate the Crucible to work so well.
Kaidan may survive it, but the biotic implants are directly connected to his nervous system. That's the reason he still has his L2s and hasn't replaced them with better ones to get rid of his migraines. Nobody, even in that day and age, dares to replace those things as there's a huge risk it could easily cripple the patient. My point is, if Kaidan's implants got damaged or, goddess forbid, destroyed, that would very likely injure him really bad and possibly leave him with permanent damage. It's not unlikely he could get paralysed for the rest of his life or something. That's at least what the game has to say about it. However, we see him get out of the Normandy just fine.
It could be that the Catalyst is just trying to be a sneaky little bastard as you say. That conversation would make much more sense if that were the case. But I'm sincerely afraid that is not what the writers intended. Always good for a headcanon, though. I mean, who can say that's not true, right?
I'm all for more tech getting destroyed with low or medium EMS. Variation is good. But perhaps that's also why I feel like leaving EDI and the geth out should be an option with very high EMS. Especially since we don't really know why "all synthetics" must get destroyed and what it means. (Again, me having too many questions perhaps.) That just gets thrown around like a fact without the Catalyst getting into it. So, it's not just EDI and the geth, which pretty much destroys the "they have reaper parts" argument, but "all synthetics". I mean, there are three version of destroy and yet there wasn't space to give our synthetic friends and allies a chance. And I can't just let the Reapers live. I find that way too risky. And Synthesis is not an option to me. But that's just me anyway.
I don't think saving the synthetics would really fix the ending for me as my biggest problem lies within the existence of the Catalyst itself and the conversation I have to have with it, but I still wish I at least knew why "all synthetics + some hardware, but not a lot of other hardware". Ahh, I'm making my head hurt, lol.