Like with everything else, I hope my posts are helpful because as another pet owner, what I am doing might be different from what you are doing, and hopefully you will get some useful ideas from them.
I get what you're saying about what to do if she's suppose to be in a pen because you also want to bond. The pen is a training tool, and things will change! Eventually she will spend less, and less time in the pen.
How I used the pen with my old man Kaji (RIP) and how I plan to use it with Ijak, is like I mentioned before, he will stay in the pen while he cannot be watched. When I will let him out, I will open the door, pick him up and IMMEDIATELY take him to his designated potty area. That can be a piddle pad or, as in my case our backyard. If I pick him up, he can't have an accident while we get to the designated potty area. I'll give him a 15-20 minute potty break. Puppies can get distracted so easily, so it's important to give them a chance to sniff, and actually potty. Of course, I will be encouraging him the entire time. If in the allotted time he doesn't potty, back in the pen he goes for 20 minutes and then we'll try again. If he does potty in the right place, and I can commit 100% of my attention on him, then I will let him roam, or I will engage him in play/training. By watching him like a hawk, I can correct unwanted behavior. For example, if he were to lift his leg to mark, I would distract with a noise (clap, or a loud "nuh uh") then redirect to a more positive behavior. At least that's the plan! It worked with Kaji, but he was already an old man when he came home. Ijak will be 13 weeks. I hope it works with him too!
As for the rug, can you move it temporarily? If she's using just the one rug it may be best to move it until she's fully potty trained. (I'm assuming your home isn't full carpet) Also, try to really clean it. I've had luck with Nature's miracle to remove pee smells.
As for the piddle pads, I am stealing this from the puppy book provided by my breeder:
As for your issue with the cat... sorry! I'm no help. I'm allergic to cats, and Kaji passed away a couple of months ago. My plan of having 2 pets at once didn't work out. I'm sure others can chime in that have a yorkie and a cat at the same time.