A Center Controller reported an air carrier encountered pressurization failure and descended to 10;000 feet.

Narrative

ZZZ [ARTCC] Sector X called to inform me at ZZZ1 [ARTCC] Sector Y; Aircraft X was returning to ZZZ due to an unexplained loud bang from the back of the aircraft and they had not required special handing at that time. On check on it was confirmed they were not [requesting priority handling] and intentions were to proceed on course to ZZZ. A few moments went by and Aircraft X transmitted something unintelligible; I asked to say again with no response. Shortly after Aircraft X requests priority descent and I see them leaving FL270. At this point I did not know if it was a pressurization issue or flight control issue. I PVD'd (Plan View Display) them to ZZZ [ARTCC] Sector Z below the aircraft while telling Aircraft X ZZZ1 was about 30 NM East of their position and gave clearance to 10 thousand with the altimeter. I alerted the supervisor I had a [priority handling]. ZZZ [ARTCC] Sector A; an adjacent airspace beneath Sector Y; over heard what was going on and quick looked me. A d-side arrived to help me with the [priority handling]. ZZZ [ARTCC] Sector Z called Sector Y to alert me there was an aircraft in confliction with Aircraft X; Aircraft Y. Either the d-side or another answered the line to let them know Aircraft X was executing an priority descent. As I was attempting to find out more about the situation from the crew of Aircraft X; conflict alert showed me the confliction with Aircraft Y. The d-side told me ZZZ [ARTCC] was expediting Aircraft Y down and turning them right. The targets appeared likely to merge so I instructed Aircraft X to turn right immediately to avoid the conflict. As I was issuing the turn to the right; I saw XXL go into the datablock. I was confused why ZZZ [ARTCC] controller would attempt to expedite Aircraft Y descent when I didn't think you could out descend a [priority handling] descent. The XXL in the datablock was contrary to what I was told the aircraft was doing. I told Aircraft X to disregard the right turn and turn left heading 280 immediately and gave a traffic alert to Aircraft X about Aircraft Y. Fortunately Aircraft X was through Aircraft Y's altitude by the time the aircraft turned. Separation was not lost with Aircraft Y. As the aircraft leveled at 10 thousand I confirmed the aircraft was stable and verified the nature of the [priority handling] as a loss of pressurization. I verified Sector B; the airspace that owned 10 thousand was watching Aircraft X and we had control to clear the aircraft direct ZZZ. Aircraft X then [advised ATC]. I asked if they had some other [urgent issue]. They replied negative; I assume they were just running a checklist; and let them know we had already [gotten priority handling] for them and reissued the original squawk code. We asked Aircraft X for the number of people on board and fuel; ran our [priority handling] checklist and coordinated with the sector ahead. I wanted to talk to ZZZ [ARTCC] myself; but not knowing why Aircraft X was descending made me choose to stick to listening to the aircraft and not getting on the landline. I could hear that someone had answered their call and let them know Aircraft X was a [priority handling]. Aircraft Y did turn right and XXL was put in the data block which seems like just a 'heat of the moment' error. Also hearing they were expediting Aircraft Y down did not seem like the best choice. These well intended action caused confusion and second guessing my instructions leading to a delay in separating the aircraft laterally. Luckily vertical separation was achieved before the aircraft got close. Sector Y D-side could not coordinate with Sector B because a line to them is not mapped on the VSCS (Voice Switching And Control System) . Commonly; aircraft with pressurization issues go to 10;000 feet because that is what they are trained for. The most common altitude approach controls go up to is 10;000 feet. It's very likely that anyone having a pressurization issue is going to need coordination with an underlying approach control and therefore high altitude sectors should have lines to underlying approach controls available to them.In the middle of the [priority handling] situation; I was sequencing for ZZZ on ZZZZZ STAR. ZZZ1 [ARTCC] passes back to ZZZ [ARTCC] for their two Sector Y streams; on ZZZZZ STAR enter ZZZ1 [ARTCC] AOB (at or above) FL330 and 20 MIT (Miles in Trail) (10 MIT no comp); meaning if only 10 MIT if there is no aircraft on the other stream. Just prior to the [priority handling] they gave me Aircraft Z and Aircraft A about 8 in trail when there was Aircraft B on the other stream and was #2 in that sequence. I had 3 ZZZ aircraft 8 MIT when I am supposed to feed the next sector 10 MIT to allow for compression and internal releases. Later on and while still talking to the [priority handling] aircraft; Aircraft C came to me on the ZZZZZ1 STAR. Usually ZZZ1 [ARTCC] puts up with this lack of compliance from ZZZ [ARTCC] because they have some toxic personalities that begin landline fights or become vengeful and it is just easier to fix it yourself. This [priority handling] situation is a prime example of why ZZZ [ARTCC] must be compliant with their restrictions. Route changes; vectors for in-trail and sequencing aircraft that should have already been sequenced added complexity to an already complex [priority handling] situation. I will not put up with it anymore.