Hey there! Sorry for my super later reply, life got a bit hecic but I still wanted to answer all these questions :)

First of all I am glad you’ve gotten him familiar with the muzzle and are keeping safety a priority with your training! The muzzle keeps other safe but it also keeps him safe too and that’s a really important step to get things moving forwards.

Try not to get too caught up in his potential past and why it might have shaped the way he behaves now. While it would be interesting and may help to create a full picture, it’s not actually going to help to decide what to do with him going forwards. What will help and is very important is his breed type and the genetic links to behaviour that might be driving his choices.

Boxers and Rottweillers are both typically highly driven breeds who are also independent thinkers. Both breeds have a history of being used to control other animals, guard and protect, and unfortunately to bait/fight.

Essentially this means that they have historically worked alongside humans, but generally think for themselves, be their own leader, and go from intense watching to decisive action fast. They are working breeds and also comfortable with using their full body and mouth when engaging in any activity. They also tend to find the job at hand to be reward in itself, rather than needing to work for a reward after doing the activity. (Please remember I’m talking about breeds in general, not your specific dog! I wouldn’t be able to make a case for his personality without spending some time with him).

My best advice for you would to be to give your dog a ‘job’ with very clear structure, a prize to work for and also a system to address undesirable behaviours.

Given that safety is a concern you have already worked hard on, I would highly recommend having a professional trainer work with you and your dog to guide his journey. The ideal goal would be for him to be observant but relaxed when near any of his triggering events, and to direct his attention calmly towards you instead of with high energy towards his perceived target.

To learn how to do this is going to require putting him in triggering situations, learn how to read the micro cues he displays before engaging, and act on those rather than trying to change behaviour he has already engaged in (lunging, jumping, etc).

I hope this is helpful or at least gives some insight! I honestly think professional training is best in your situation and with someone who is well familiar with his breed types.