The planet is 10 Earth-Radii, with a 1 Earth Gravity value. It is a habitable planet. It has a thin atmosphere. It also has 3 moons and a ring system.
Apart from the whole "habitable" and "thin atmosphere" thing, you've basically described Saturn. Saturn is ~95x Earth's mass, and 9.5x Earth's radius at its equator. Surface gravity is a bit over 1g, too. Even has a similar number of big moons, and a ring system.
Your requirements match a gas giant just fine, but not a planet with anything approximating an inhabitable surface. Don't be so greedy: you don't need a world that big in order for it to be interesting!
You might be able to have portions of the atmosphere be minimally hostile, with human-compatible temperatures and pressures, but it almost certainly can't be breathable. You can have floating space habitats in such a place where you could go outside on deck in shirtsleeves, so long as you brought breathing gas with you.
The moons are rather large. The largest one is the size of Earth, and Habitable. The second one is a 1/2 Earth radii, and the smallest one is a 1/4 Earth radii.
This isn't totally implausible, but it is stretching plausibility a little. That's a lot of pretty big moons. The one Earth-sized moon is probably just fine. You might have issues with its placement around the parent gas giant, because if Jupiter is anything to go by gas giant's can have very, very strong magnetic fields and associated deadly radiation belts. I'm not sure how best to solve that issue though. I'm not sure if the gas giant has to have a super powerful magnetic field, but it seems likely.
Is this system possible? If not, than why, or what can I do to make it possible?
You have two choices here. One is to be very light on the details, and declare by fiat that it is possible. You'll get away with it just fine.
On the other hand, if you start talking about distances to other worlds, and day and year lengths and all the rest, you start needing to cobble together a working system and that may or may not be practical.
I can recommend hunting down some sort of gravity simulator and fighting with it a bit to see what works. There are quite a lot of complicated factors that fit into these things... n-body spheres of influence and orbital resonances and things, more than I understand or can explain. Start with something simple, see what you can get away with adding.