In order to reach all but 2 of the Rings, you must be able to Soar using the Eon Flute. Steven does not give the Eon Flute to you until Team Aqua/Magma releases Kyogre/Groudon (respectively) from the seafloor cavern. At this point, you would have access to all the Mirage Spot Ring Encounters, provided you meet their other requirements. You do not have to capture/defeat Kyogre/Groudon to access these events. However, if you are leveled on par with the game content to this point, you will be fighting uphill, as Kyogre/Groudon are at level 45, while all Legendary Encounters are at level 50.
Heatran is a bit of an exception to this. In order to reach the Ring in the depths of the Scorched Slab, you must first complete the Kyogre/Groudon encounter beneath Sootopolis. You will encounter Gym Leader Flannery, who mentions looking for a very rare Pokemon in the area, but that she was unsuccessful.
Finally, while you can reach the Sea Mauville any time after you have received the HM and Gym Badge for Surf, you will not be able to being the side-quest associated with Ho-oh/Lugia until after the Kyogre/Groudon encounter beneath Sootopolis. The result of the side-quest will be the Clear Bell for Ho-oh in Omega Ruby, or the Tide Bell for Lugia in Alpha Sapphire. After obtaining your version's Bell, you can return to the Sea Mauville, locate your version's Ring, and begin your Encounter with Ho-oh/Lugia.