First of all, thank you all for your answers, your help is much appreciated!
I'm sorry for the slow reply, I got sucked in a work wormhole, and since this repair is not urgent it was put on the back-burner.
I took a few pictures. On the first two (overview and caps) you can see how the crapped caps looked like when I removed them as well as an overview of the board region.
Then the interesting ones, two for each pair. Both come out from MAX4477 opamps as expected. C134 and C137 then go to C100 and C102 of the second picture (the ceramic filters I talked about) which go directly to the output jack socket. For pair 2 everything arrives from the left of the second picture which is 90° rotated compared to the 3 others, but same deal ends in C99 and C104 then down to the jack below.
Meanwhile I met some guy well versed in audio gear repair during some gig, and he told me to replace the electrolytics by film ones (non polarized) as it could eliminate the issue with the negative voltage I'm measuring altogether (as well as improving on the surely dodgy factory ones)
He also told me to inject or play 1kHz sine waves and check if the opamps outputs look correct or not, which I still have to do.
To answer some of your questions: I don't think it was a phantom power issue, but during the last operation where this card was used, it was receiving LTC timecode from a console in order to slave some video playback for the show. Both were plugged on the same outlets in the same technical area, but there were tons of other stage equipment (pyrotechnics, lasers, sound and lighting etc) and the driver froze down mid-show as well as our video playback as you can imagine (very embarrassing moment) so we can't indeed exclude that some potential difference or other bad stuff happened via the inputs.
Technically the outputs were not in use (not plugged to anything) but since it was using the ASIO driver, you always have both input and output active even if you feed the output buffer with zeroes.