@martant I'm thinking don't update yet. Try to get Opera fixed before you update your OS and get stuck with it.
What I would try is to click Opera in the top bar while it's open.
I click Opera>About Opera
It then takes you to where it tells you which version you are running. On my iMac it was Version 30 something. All I did was click on Relaunch and it updated to like version 45. This still wans't the latest version when I went back to Opera>About Opera so it asked me to relaunch again Then it updated me to version 55. After the relaunch I opened About Opera again and it said relaunch again so I did and then I was on version 75. What you can do if this works for you is stop when you get to the versions 50+ if you are saying that is all that will work on Mavericks.
Like I said my iMac is on El Capitan and version 75 still works fine. I read in the Opera Support Documentation that it is recomended that all users update to the latest version of Opera so I wasn't aware of certain versions being only available for certain Operating Systems. I was looking for the documentation on which version is for Mavericks to help you out; I searched all night and couldn't find this information.