A nuclear spectroscopy experiment was conducted to study α-decay chains stemming from isotopes of flerovium (element Z=114). An upgraded TASISpec decay station was placed behind the gas-filled separator TASCA at the GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt, Germany. The fusion-evaporation reactions ^{48}Ca+^{242}Pu and ^{48}Ca+^{244}Pu provided a total of 32 flerovium-candidate decay chains, of which two and eleven were firmly assigned to ^{286}Fl and ^{288}Fl, respectively. A prompt coincidence between a 9.60(1)-MeV α particle event and a 0.36(1)-MeV conversion electron marked the first observation of an excited state in an even-even isotope of the heaviest man-made elements, namely ^{282}Cn. Spectroscopy of ^{288}Fl decay chains fixed Q_{α}=10.06(1)  MeV. In one case, a Q_{α}=9.46(1)-MeV decay from ^{284}Cn into ^{280}Ds was observed, with ^{280}Ds fissioning after only 518  μs. The impact of these findings, aggregated with existing data on decay chains of ^{286,288}Fl, on the size of an anticipated shell gap at proton number Z=114 is discussed in light of predictions from two beyond-mean-field calculations, which take into account triaxial deformation.