11 mth old rescue - training

I'm guessing a lot of people may disagree with my decision but I have rescued a beautiful 11 mth GSD and I live in an apartment. I have had her for about a month. My girl gets a great deal of exercise ie. 2.5 - 5 hours a day. She does get time off leash and luckily I have friends/family with dogs and fenced backyards. I also take her in the park behind our building and play fetch with a 50 ft leash.

I have 2 training issues to ask about: barking - I know I could have trained her to bark/quiet with positive reinforcement. However, I found this confusing and the timing difficult. I was also told I could let her bark 3 times and then correct the behavior - I also found this difficult and confusing. So I chose to correct- period. I tell her no and if she is heading toward the sound ie. someone passing by the door I head her off and direct her back to where she was, at this point if she has not already stopped I make her "down" and then she stops. I only do this when she barks at things such as the neighbours passing by, I don't correct her for barking when people come to our door. As soon as she is quiet, I say "quiet" and give treats/praise for that behavior. I think this process is dominant/intimidation and I know I have chosen the easy way out but it does work and I do have to be respectful of my neighbours. In some ways I think it is less harsh than an e-collar but do you think this could negatively effect our relationship?

2nd issue - our cat. I was told she was socialized with cats however, the first time she saw our cat she went nuts. She was in her crate and she barked and lunged. We have kept them separate and bring them together for supervised visits (a trainer suggested this strategy). DH and I correct any barking (over-excitement/aggression) towards the cat and give treats when she is calm. However, the cat is also aggressive towards her and when she is allowed to sniff the cat - the cat growls. We always hold the cat and try to calm it as well. The cats behaviour leads to more excitement/fear/aggression? In all of our attempts everything is ok until the cat reacts and this triggers the dog. My question is: do you think I should just let the cat have her home back and keep the dog on a leash in the house, correcting her for every negative behaviour toward the cat (and rewarding the positive), do I just continue with supervised visits hoping things will improve, or is there some other solution? The trainer I talked to said that it is important not to create anxiety with the presence of the cat. That is why she felt the cat should not be out walking around in front of the dog until there was a positive association. I just don't know when it is time to help the dog understand that the cat lives here too. I am also aware that some dogs are aggressive towards cats and never learn but I seriously hope that is not the case. Anyway, thank you for any input. Sorry for the length.