Reaper is just as susceptible to Boom, Headshot! as anyone but...it's a robot? It can also swim with no trouble and apparently automatically heal damage.
The healing bit is explained in the in-game wiki actually. Apparently, all ground based unmanned weapons by 2065 use self-repairing smart material in their armour that grants them a limited ability to "heal" from damage; Reaper, being an advanced prototype, has a superior version of this. The swimming bit is likely a result of of said prototype status as well; Reaper is probably fairly lightweight and likely has some sort of ballast tank inside of it to give it buoyancy. The head shot thing is probably just for balance reasons.
Maybe that's where his (its?) processor is. A computer can't function with its processor destroyed, what more a robot?
Yeah but that leads to the question of "Why does the super-advanced combat robot play Cranial Processing Unit straight but all the other mass-production robots in the game avert it?"
PerhapsSuper Prototype still means "prototype" and, while he features amazing advancements, they still have some design flaws in other areas?
Or perhaps his chassis is already full. After all, he has to store all that ammo for his minigun arm somewhere and he doesn't have a visible ammo backpack, not to mention being an advanced prototype designed to support special forces units in the field means he probably has an oversized power unit. Add in the above mentioned air bladder for underwater use and his chassis is probably full. Not to mention that Cranial Processing Unit isn't as bad an idea as its detractors make it out to be; after all, soldiers are trained to aim for centre mass and a head can easily be removed and placed on a new body if the original is destroyed.
The writing goofs when you're playing as the female protagonist. The storyline is clearly biased towards the male, with Ship Tease between the Player and Kane regardless of gender, and occasional moments where the Player is referred to as "he". This is because you are reliving the memories of John Taylor's previous mission, who was decidedly male and was in a relationship with Kane.
There are two separate sections in the game where the level's current antagonist takes control of G.I.U. robots and survives, despite Diaz telling you that doing so will fry your brain due to G.I.Us operating via a collective hive. Since Diaz and Taylor are both connected to and controlled by Corvus, a Hive Mind, it stands to reason that they should be able to control G.I.Us without incident.