Big Daddy A Cyborg :
A Big Daddy is a Cyborg (part human, part machine joined (grafted together)) :
We assume they did use the genetic and medical tech that their genius created to solve several significant problems.
For the grafting - tissue/foreign body rejection issues for attached mechanisms - the treatment/whatever that keeps their tissue from reacting significantly. Not sure to what extent the 'grafting' is done. Structural reinforcements, control mechnisms. Plasmid/Tonics could manage this - preferably long lasting and 'stable'.
For the Diving Suit problems (we've seen the Big Daddies out in the ocean) - those are not hardsuits that keep the pressure out, they are the conventional flexible diving suit (real ones are rubber sandwiched between cotton twill canvas). Other stronger 'clothish' materials might be used, but can't be too stiff, and need to resist salt-water corrosion.
In Real Diving Suits of that type the air pumped from the surface was kept slightly higher pressure than the depth's water pressure (so the air wasn't squeezed out of the helmet and the body pushed into it). The interior of the suit always is at a slightly higher pressure than the water's depth. SO you pressurize to the inside to just over the outside water pressure (around 19 atmospheres of pressure at only 600 feet -- that is well beyond safe/survivable diving (which is 150-200max for humans -- where Nitrogen narcolepsy, nitrogen narcosis were the hazards). You have to keep adjusting the pressure as the depth changes. "The Bends" (google it) could happen when the Daddy comes back inside and the pressure is quickly lowered to the 1 atmosphere inside Rapture - re-entering Rapture (via an airlock). You also get the pressure change effect simply moving between different depths. I would not put Rapture too deep, or the materials/technical problem increase geometrically, beyond any practicality.
So the Daddy needs some Tonic/Plasmid to sweep gases out of its blood rapidly for that frequent pressure transition (or to mitigate the effects some other way) to prevent the bubbles shutting down its vascular system. A Big Daddy could potentially work at much greater depths.
Then there is the strength to move in that very heavy suit. The BD suits are heavy and the oversized helmets and devices on the back which real human divers didn't have (air supply/pressure pumps was thru air tube from surface/elsewhere...) and add even more weight from adding tools/weapons, plus tanks and their power systems and those heavy weights (that keep it from floating by displacement buoyancy). It all would weigh MANY hundreds of pounds (600-800 total weight from what the displacement (volume) looks to be. That would require Strength tonics (we've seen 'Brute' tonics exist...) to super grow muscles and bones to support/move that weight. They would require even more strength to carry all that outside of the water.
To power that physiology, their digestive/vascular/etc.. systems would also have to be enlarged/made more efficient - bulky/bigger size means squeezing into the suit (closely or the suit gets bigger/heavier and the body needs to in turn carry that, requiring even more weight then more strength, etc...)
Add whatever mental conditioning (possibly reinforced via Plasmid/Tonic/other) needed for the Big Daddy to operate/behave as it does (still capable of recognizing problems and solutions, handling tools to make repairs that all takes coordinated actions and planning etc ) and to take direction for tasks or act independantly to a limited extent.
The technical part of the suit has to hold enough oxygen for a normal work cycle (time to get outside/walk to work location/do some work/get back to wherever to recharge - preferably more than 30 minutes of work each cycle). With a larger more energy burning physiology, perhaps some Tonic/Plasmid (or other) helps with that to make the respiration more efficient.
I haven't figured out if there is any 'power assist' for the main body actions. Combustion - I'm not sure can work at that pressure - 20+ atmospheres (and it would require oxidizer be carried to function), so batteries and electrics might be the only workable solution - possible too limited so that it would be only for the tools).
The Big Daddy's damage resistance might be similar to the Brute's, but with the thicker covering and some projectiles bouncing off the various metal bits (covering 1/3 the surface?) it can absorb more damage. Fortunately patching the suit is easy, as it doesn't have to hold more pressure than a few PSI (pounds per square inch).
It is possible that the Big Daddies suits are liquid filled (and they breath a oxygenated liquid - ?? that yellow stuff in the big daddy factory in BS1 ???). That kind of mechanism lowers compression/decompression effects somewhat, but having fatal bubbles form in your blood/tissues is still there with the kinds of depths involved). SO the Plasmid to take care of that would still be a needed solution - possibly to work to very great depths (sea depths OFF Raptures's underwater volcano go to 10,000 feet deep, and more - - - 300+ atmospheres of pressure ).
What routine repairs/replenishment do Big Daddies get (and from where)? All will probably be explained in the MMORPG.
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Delta's Potential for Surviving :
Not sure if the Daddy-Lil Sister link being broken would have been the cause of Delta's death (like what happens if it is reestablished - seemed to be a combination of mental conditioning and physiological).
That final explosion may have severly damaged Delta (the hamburger making effect of powerful blast compression wave - it might have been more realistic to have his suit appearing well mangled afterward) - and his body damage beyond the capacity of Med-Kits to repair.
At the end of BS2, our little group was out in the Ocean and they went upwards quite fast, so allegedly were not close to any Vita-Chamber (WHY NOT one in the 'LifeBoat' ? -- we found them in every other strange place previously....), and it would take a long time to get back near one (we're not given any data on how long you have for that 'resurrection' system to do its work....)
It is actually possible that as Delta 'died' and floated back down to Rapture to have had a Vita-Chamber intercept him. It is all largely unexplained how that Vita-Chamber mechanism really worked -- Flying Monkeys/Lock Box Gnomes carrying/dragging your still warm corpse into it, to be repaired. That is a bit easier to explain than real 'Teleporting' ( or worse, Delta's being reconstructing entirely from nothing but DNA and INCLUDING his diving suit/equipment - which is definitely NOT part of your DNA). Unfortunately that game-play crutch (handling of player 'death') they chose to include made it part of the plot/Canon/Lore.