It sounds like you're off to a great start. Basically, you're trying to set the fundamentals in place, right now. Just keep telling yourself that no matter how much work it seems like, you're doing yourself a favor. If you instill good habits in your pup now, you won't have to continuously worry about things for the rest of his life. A few month of work now, saves years of aggravation later.
For instance, my Jack Russell was given some basic house training when young, but I made mistakes. Now, at 12 years old, I can expect pretty good behavior, but she'll still wander off into a far corner and pee occasionally if she's been in a long time. If someone is around, she asks to go out, but I can't trust her the way I can the keeshond we inherited from my grandmother. She'll hold it for 3 days if necessary. I've seen her do it when she refused to go to the bathroom on vacation, no matter how many walks I took her on.
You're doing excellent in restricting her access to the house. Many people want to give the puppy free run, like you'd eventually like to do. However, the space is so big that a puppy doesn't know yet that all of it is the "den". In the wild a dog, wolf, etc... won't want to dirty it's den, so it'll go outside and mark it's territory out there. That's what you're doing now. Right now, however, the puppy sees it's crate as the den and anywhere else as territory to be marked. Over time, the puppy will come to see the whole house as the "den", but that takes time and acclimatization.
I recommend that the puppy be locked in the crate at night, but during the day, be put in an exercise pen. These are light weight wire panels that connect together to form a playpen. You can put the puppy and a bunch of toys in here, rotating the toys periodically to avoid boredom. This pen can be placed wherever you like. You can even separate two panels and connect them to the corners of the wire crate, so that he can go between the two. This is a great place for her to be when you aren't directly controlling her and if she's trained to it as a puppy, she won't try to get out later in life, even if she gets large enough to destroy it or just jump out.
You don't want to let her loose anywhere in the house unsupervised. You'll learn over time when she can be trusted, because she'll have either had mistakes or not. Most likely she will. I'd say a 99% chance. At 9 weeks, she has no bladder or bowel controls. The muscles haven't developed yet, the same way a human baby needs diapers. The rule of thumb I heard is they can hold it for 1hr per month old they are, not to exceed 9 hours. They'll vary though, and she'll probably be able to hold it all night, well before 8 months old.
It's okay if she goes in the house and you shouldn't say anything negative to her. Simply say, "no" in a calm voice, pick her up, and take her outside if she goes. Then clean the spot with a cleaner designed to remove pet scent to keep her from wanting to go in the same spot again.
Also keep in mind that any dog is more liable to go to the bathroom after waking, eating, drinking, or playing. So be more diligent at these times.
If you keep up this routine and are diligent about it, I think she'll be roaming the house by the time she's 6 months old, maybe sooner if she learns her lessons well. You get out of it what you put into it. Some dogs are harder than others to house train. Just keep in mind it's a set of scales. The more times she does bad and you aren't there to show her the right way, she get beads in her untrustworthy pile. Every time she goes to the potty outside or even if she has an accident, but you say "no" and carry her outside, she gets on in the awesome dog pile. You can't ever remove beads from a pile, just outweigh them on the other side. So stick with it and I think you'll be happy with the results.