Air Carrier Captain reported a communications breakdown between flight crew and ground personnel; resulting in flight crew being provided with erroneous Hazmat documents. Multiple handling errors also resulted in dangerous goods and a kidney not loaded on aircraft.

Narrative

We operated under Medevac; but left the kidney in ZZZ.It all started with a call to Maintenance. While powering on the aircraft; the message 'EICAS OVHT' flashed a couple times followed by 'EICAS FAULT.' At this point; I noticed a little bit of moisture surrounding the Display Unit. I called Maintenance who suggested that I run the EICAS FAULT check. It worked; the fault cleared; and by the end of my call with Maintenance; the moisture around the Display Unit's dissipated. (I only added that because the moisture was in the back of my mind the entire flight and may have contributed to me forgetting the kidney.) Right after I ended the call; my First Officer got back from his walk around and told me that the nose wheel was missing a bonding strap. I believe it was at this point I'd take a look at the release and was aware of our Medevac status. I called Maintenance back and they told me they'd be sending Maintenance out. While waiting; we set up and briefed for the flight like normal. While briefing the departure; the mechanic arrived and took the logbook with less than 10 minutes to departure time. [Mechanic] handed it back with one minute to go. I took a quick look at it; said it was good and we closed [aircraft door]. After the before start check list; I decided to recheck the corrective action and saw that the mechanic had written main landing gear instead of nose landing gear. I called Maintenance and asked him to express urgency because we were a Medevac flight. While waiting the second time and waiting for the takeoff numbers because the plane was still showing out of service; I asked the rampers if they were sure they had loaded the kidney. He told me he'd double check and came back a moment later saying it was. At XA:55; ten minutes after departure time; the mechanic gave us the logbook back with the edit. We quickly got our takeoff numbers; pushed back and taxied out. While taxiing out; Ground asked us to confirm we were a Medevac flight. At this point; I thought we were; so for the entire flight; we kept the status of Medevac. Just before we started our descent into ZZZ1; it dawned on me that we never received a notification about Dry Ice; thinking that a human organ would need to be kept cool. I checked the takeoff remarks and didn't see anything. I messaged Dispatch who said he'd check with Ops. After we landed; I saw the message that we might not have received the kidney. After parking; I went outside to watch the ground crew unload the bags until they gave me confirmation the kidney was not on board. I did however notice a box wrapped in a red plastic bag. I believe the ZZZ rampers mistook that for the kidney we were supposed to have on board.Even though it would have all but guaranteed a delay; had I gone and looked that the kidney was onboard with my own eyes; all of this would have been avoided. The fact that I was trying to avoid any delays so I could do my part in saving a life ended up causing the biggest delay of all.