CRJ-200 Captain reported loss of cabin altitude during climb. Crew descended and continued flight.

Narrative

I was Captain of an airplane had the right pack inoperative and deferred which resulted in an unpressurized state and landing. At the gate in ZZZ; our fueler put in more fuel than our release full which was below our maximum takeoff weight which put us at risk of landing overweight. Additionally; our ramper initiated a pushback without our request after we released the brakes to him. Our flight was delayed from maintenance servicing and I believe the ground crew felt rushed and in a hurry to get us off of the gate. The ramper pushed our plane for about 10 feet before we were able to put on our headsets and tell him to stop pushing. I was disturbed from his careless mistake but told the operations there what happened; recovered; took our time; completed our briefings; checklists; told him when we were ready for push; and began the flight. We took off bleeds closed and used the procedures recommended in the MEL to preserve the remaining left pack and our pressurization while swapping the bleeds from APU to the Engine at 5;000 ft. While climbing to our planned cruising altitude of FL240; we asked ATC for delay vectors at some point during our flight because we wanted to fly for an additional 10 minutes to burn enough fuel to ensure that we landed less than 47;000 pounds to prevent an overweight landing. While on a delay vector during our climb; we received a L Pack High Temp caution message which temporarily appeared and disappeared. A Cabin Altitude caution message then appeared. I was pilot flying so I ran the Threat and Error Management model for an unanticipated threat. I verified that the airplane was still climbing in our intended direction at a proper airspeed. There were no memory items or QRC items to complete. I transferred the flight controls to my First Officer and began running the QRH procedure for Cabin Altitude Caution message. While in the first few steps; a Cabin Altitude Warning message appeared. My first officer and I both immediately donned our oxygen masks and attempted to make contact. I struggled with getting my oxygen mask microphone to transmit but my First Officer was able to help me with that. We then established verbal contact with each other stating we were on O2. The Cabin Altitude displayed with a reading of 15;000 ft and the passenger oxygen masks automatically deployed. The FO needed to run the Emergency Descent QRC. We did not do a proper transfer of controls back where I said 'My Controls' and switch the transponder and flight director back to my side; but it was understood between my FO and I that I was back to Pilot Flying in that moment because I spun a lower altitude into the control panel; started a descent; put my hands over the thrust levers; I pulled the spoiler lever to max; and told my FO to request priority handling and emergency descent. He did advise Air Traffic Control. He completed the Emergency Descent QRC while I descended all of the way down to 10;000 ft. There was a step that says PASS OXY (FO) SELECT/ON. My FO did not press the button however the button was illuminated ON with a caution message that confirmed the Pass Oxygen was On. While descending and several times during our cruise at 10;000 ft; my first officer checked with our flight attendant for the status of the cabin. Our flight attendant told us everyone was safe and she made frequent checks on each passenger. Our airplane was in a safe state at 10;000 ft. I canceled the priority handling status and told them that we did no longer needed rescue trucks and services standing by when we landed. We decided that it would be best for my FO to finish the flight as Pilot Flying and we established that and made those roles clear. We had enough fuel to continue to ZZZ1 at 10;000ft and finished the flight to ZZZ1 with a normal landing at our destination airport. Once at the gate; maintenance and the move crew arrived to take the airplane to the hangar. Due to the nature of us being task saturated and the lengthy debriefs andphone calls to our Manager on Duty; Maintenance; and Dispatch; my First Officer did not do a final walk around.Cause: Maintenance deferral of the right pack contributed to our loss of pressurization in flight. Both auto pressurization systems 1 and 2 also failed in flight.Suggestions: I suggest airplanes are not dispatched with a single pack.