Congratulations on your new dog!
I definitely would try crate training. But if she's used to eliminating in her bed, the crate isn't going to stop her, and you'll just be reinforcing that behavior unless you take her out so often that she never has a full bladder. That's your best bet, in my opinion. Just make sure to take her out VERY regularly and often to allow the most opportunity for her to go outside and receive praise. Every time she goes inside reinforces the fact that it's ok. So ideally, every elimination would be outside.
Never punish for accidents. Just swoop her up and take her out. Then as she's going potty outside, calmly say, "good potty" (or whatever phrase you choose) and give her extreme praise and treats immediately afterward. If she goes in her crate, be sure to give it a thorough cleaning to remove ALL odor.
Introducing a New Dog to a Resident Cat
I would be real careful since she's probably not house trained at all. She probably has no reason to believe that she shouldn't go inside, so it's going to take some real attention on your part to train her. Watch her every minute. It's not the amount of hours you need to be concerned about, it's the simple fact that she has no reason not to think it's not perfectly fine to pee wherever she's standing.
I definitely would try crate training. But if she's used to eliminating in her bed, the crate isn't going to stop her, and you'll just be reinforcing that behavior unless you take her out so often that she never has a full bladder. That's your best bet, in my opinion. Just make sure to take her out VERY regularly and often to allow the most opportunity for her to go outside and receive praise. Every time she goes inside reinforces the fact that it's ok. So ideally, every elimination would be outside.
Never punish for accidents. Just swoop her up and take her out. Then as she's going potty outside, calmly say, "good potty" (or whatever phrase you choose) and give her extreme praise and treats immediately afterward. If she goes in her crate, be sure to give it a thorough cleaning to remove ALL odor.
It depends on what she's used to, if anything at all. I would just see how she responds to a collar. If she pulls, you'll just have to work on walking on a loose leash. If she can walk on a loose leash, it doesn't really matter what you use.