If we extract the first molar (#6 marked with red), will the next molars (#7 & #8) move to the the space of first molar? (photo)

Sounds like a simple solution but it does not work well as an adult, more likely to tip forward creating future issues.  A tooth can be moved forard with braces but still difficult.First choice if possible restore tooth with root canal. Second choice extract and palce implant or a 3 unit bridge.We used to transplant teeth 30 years ago where you could extract the first molar and transplant wisdom tooth into socket.  Implants are now more predictable.  Best time to transplat was before complete formation of the root.  Ira L Shapira DDS, D,ABDSM, D,AAPM, FICCMOThinkBetterLife.com

The short answer is that it is highly unlikely that the teeth on the right side will move into place after the extraction of the root canal tooth. You were 13 when the extraction was done on the left side and your 2nd molar (#7), the "12" year molar, was just erupting and filled the space. Even, so it had to tilt to fill the space. The root canal tooth on the right side appears non-restorable and should be replaced with an implant and crown. Seek a consultation with an oral surgeon or periodontist and general dentist to assure that there are no sinus issues as far as an implant is concerned. Also, the surgeon can elect to remove #8 on the right side. Best wishes!