Welcome to the forum, and thank you for doing some research before posting! It's a rare find nowadays
I don't want to necessarily steer you away from a rescue, but I feel like I need to share my story about adopting a rescue since our situations are similar in many regards.
My hubby and bought our first home after we got married, and I made sure it had a fenced in yard since the plan all along was that we'd get a dog once we moved from the apartment into a house. We wanted a larger, protection-ish dog so (like you) we had to decide between a Doberman and a GSD. We ended up choosing the Doberman for no other reason than the hair factor!
We had our two 4 year old cats that we've had since kittens, and they were our kids. Like you, I'm more of a cat person but have grown up with family dogs and wanted a dog of my own finally.
After doing some research, I thought adopting a rescue would fit our lifestyle better since we wouldn't have to worry about housebreaking, etc. So I found a Doberman rescue and we went to look at the place and the dogs one day. We ended up choosing a 2 year old male who was very sweet and who had apparently been fostered with cats and did very well. Getting along with the cats was my first priority since I would never put them in danger.
He was quiet as a mouse on the way home and seemed like a very calm, well-adjusted dog. We were so happy! We got him home and I kept him on a leash just in case. The moment he saw the first cat, he zeroed into prey-kill mode and lunged. Luckily I had the leash on him so no one was hurt, but he clearly would've killed the cat if he had the chance.
Thinking maybe it was just since he had never met these cats before and it was a new environment, I didn't think much of it. But my cats had never really been around a dog before, so I was afraid they would be traumatized from almost being attacked.
All day, the dog laid by our feet and didn't move. He was a very calm and nice dog, but whenever he'd spot one of the cats, it was back to kill-mode.
Fast forward to that night, we put him in his crate in the room across the our bedroom (our house is pretty small so we could still see us in bed from where he was). He went completely ballistic ALL NIGHT. Barking and howling and trying to claw his way out. It was horrible, but I couldn't let him free for fear of him finding one of the cats.
Sleep deprived from being up all night, we both went to work in the morning. We crated him again and when hubby came home on his lunch break to check on things, the dog had mangled the bottom of his crate (mind you, this was a large, sturdy wire crate), dug up the brand new carpet and padding under the crate, chewed the bars and managed to weasel out of the crate door somehow. There was blood everywhere from him digging and chewing so frantically. There was a 3 ft x 3ft hole down to the plywood where he completely trashed the carpet trying to dig his way out. Luckily we had shut the door to that bedroom, so even though he got out, he wasn't able to get out of that bedroom and get to the cats.
I felt bad for the dog, since he obviously had horrible separate anxiety - who knows what had happened to him in his previous life - so we decided to give him another night. It was a repeat of the first, and worse.
The day after that, we decided we could not keeping doing this. I drove him back to the rescue, sobbing uncontrollably the entire way. This poor dog being shipped around place to place, and who knows what would happen to him now. I explained to the rescue that he tried to kill our cats and had extreme separation anxiety and we could not keep him. I felt like a horrible person, but I knew we couldn't handle this dog. Maybe if we didn't have any other animals and one of us didn't have to leave to go to work, but that was not the case for us.
That ordeal almost put us off owning a dog forever. Six months passed and we decided maybe if we got a puppy and could raise him around the cats, it would be fine. I did a ton of research on reputable breeders (this was even before I joined this forum), and found one in my area. He was expensive, but I knew the difference, even then, between BYBs and reputable breeders and he was worth every penny plus some.
Fast forward, we got the best puppy in the world (sorry everyone else but he is!). Red was very smart and simple to housebreak, was fine being crate trained (though he earned free run of the house by 1 year old), and is completely fine with the cats. We even added another cat, who absolutely loves Red and is always rubbing up on him and playing with him.
I'm not anti-rescue, on the contrary really since there are so many nice homeless dogs out there who just fell on bad times (like situations that MaryAndDobes mentioned) - but I just want you to know that some dogs may come with issues, and there's no way of knowing which ones until you get them home. I would be especially careful if you have other pets, especially cats. It only takes one mistake and you've got a dead cat.
That being said, there are countless people who have adopted rescues who have been completely perfect with no problems with other animals. They are lucky because I wish it would've worked out that way for us. Although, I'm actually glad it didn't because then we would've missed out on Red, who will never be replaceable in a million years.
Good luck with finding a dog. You sound like you will be a great owner, regardless of the breed you choose.