Can I make E-7 with a bad history over 10 years ago?

Yes you can. It will be a while but you got to keep grinding hard and do the best you can at all times. It took me 5 years to overcome a GOMOR and bad NCOER due to a DUI charge that was i not guilty off. But they still slapped me with GOMAR and did not meet standard Ncoer regardless of the court outcome. I went through the QMP process also and survived that. The Army gave me another chance and I have not been wasting it. I received 4 great NCOERs since then. I got passed up for MSG 3 times. Had to be because the GOMOR was in permanent file but last year I got it moved to restricted file and I got picked up last month for MSG. I know they saw the bad NCOER I recieved about 5 years ago but I think the board members also see if your a resilient Soldier that can bounce back and excel after a possible career ending movement. Keep furthering your civilian education and take as many military courses as you can. BattleStaff etc. That with great NCOERs will prove that you are deserving and you truly learned from your mistake.

It will not be easy, you will have your best look after the year mark on DI status through to tour completion. Be patient and do well. Thank you for your service.

Yes - you've done all the required things to show you should be retained. You're not first to screw up. Do your job and try to do it well enough so your current or future present GO will present you with an award.
You're talking 10 years ago and surely you haven't been supervising extra duty all that time?
Any combat awards or commendation awards since then - a good conduct medal outshines a letter of reprimand. Especially one auto-generated because you violated policy.
Additionally you have had more successful NCOERs since that date.

It'll be tough, but it is possible.

Look up the board AAR for your MOS when it comes out. Find what got the highest percentage and make sure your in that bracket next year.

It is definitely possible. I've seen it happen. Fill every possible square, get your degree, do all your PME, etc

In my opinion, I feel you can still make E7. The Gomar has been moved to your restricted file (which it takes awhile going through that process to do) You already beat QMP and you are a Drill sergeant. Your chances at getting picked up are as high as your other peers competing yearly. Keep going after the hard assignments and hopefully you pick up this year. If not, just know you are still in good shape to make it.

Similar circumstances. Completing a 3 year "B" billet ("B" billets are DI, Recruiter or Embassy) made the difference. If you are still being sent to schools by your command you are likely identified as having potential. Otherwise they wouldn't waste the resources.
Keep your head up and good luck.

Have you considered writing a letter to the president of the board?

Honestly, you're probably only going to make it if they promote a bunch of people or someone on the board knows you and maxes you out. It's unfortunate, but the best advice you can give to someone that gets into trouble early in their career is to ETS. The zero defect army is alive and well, and it's nearly (not completely) impossible to overcome any type of bad paper. Think about who's on the board. They promote people in their image. You're much more likely to make rank as a mediocre performer that keeps their nose clean than as a top performer with any disciplinary history, regardless of how long ago it was. Wish you the best, but don't get your hopes up.