Sounds like alert barking and demand barking, which are two very different things and can be addressed separately.
Peggy was the ultimate demand barker and I addressed it in two ways:
1. I gave her whatever she wanted the moment she stopped barking. In my typically human way, I wanted to punish for the barking by withholding the object of her desire. But her brain wasn't making the connection. So I let her bark, just ignoring her completely (even while it occasionally escalated to a torturous degree), and the second it stopped: REWARD. No exceptions. She eventually figured it out.
The cat likely won't be onboard with this plan, so hoping some cat owners have some cat-specific ideas to share.
2. I stopped pushing her patience. I think every dog owner loves to watch their dog sit (and sit....and sit...) for every single thing they want. But don't push it at this age. Your puppy is still learning impulse control. So: Puppy wants a toy? Sure, ask for a sit. You might get a bark or two. Just ignore that. But the moment that little bum hits the floor, send that toy flying with great excitement. Don't pause for even a second. Really rely on your reflexes here. Peggy caught on fast, but even at 16 months, she occasionally needs a little tune-up.
For alert barking, I try to honour Peggy's instincts. Even when it seems like she's barking at nothing, there's definitely something that's triggered it. So I go over to the window, check for danger, thank her and tell her she did a great job. Now let's go relax.
Some days she's more alerty than others. (During her second heat, I almost lost my mind.) But not giving in to my human impulse to shout "NO!!" noticeably reduces my stress level.
For barking when walking out the door, you can try what I did: Arm yourself with yummy things and redirect back to you (position treat directly in front of puppy nose, then deliberately draw a semi circle in the air back towards you). Reward for puppy following the treat and offering silence. Then repeat, repeat, repeat, as much as necessary, never allowing a step forward while barking.
Sometimes it takes until the end of the driveway to get Peggy tuned into me. But I no longer lure her with the treat. It "appears" when she voluntarily quiets herself and looks to me.
You may even be able to solve this particular issue faster by simply leading the way out the door. An adolescent dog charging out the door ahead of the human is recipe for a barkfest.
Just try to remember that even though we just hear BARKBARKBARKBARK, a bark has lots of different meanings, so it's helpful to address them as the unique messages they are:
Is my puppy excited? Frustrated? Anxious? Bored?