From a medical perspective, the only thing I see wrong here is the first time being sent home without a workup or an ultrasound or anything. They doctor pushing on your uterus. . . he needed to examine it, to feel size, feel for contractions, etc.
I don't understand why you think you "made" the hospital admit you. . . please explain. I can't imagine a hospital not taking action on a bleeding pregnant woman. When did they decide to send you home? Had you stopped bleeding? Had there been confirmation of your baby's heartbeat before you left? What exactly abnormal was found in the ultrasound?
Without more detail it's impossible to tell if the hospital was negligent or not. Patients have a very one-sided view of their care and don't always understand exactly what's happening behind the scenes and what certain conditions warrant which kinds of actions. And to be honest, at less than 23 weeks gestation when a baby isn't viable, they simply do not take heroic measures to save babies - in many cases if complications develop that early you really can't stop them or prevent delivery anyway.
I am terribly sorry for your loss, but often things in pregnancy can go wrong quickly, with little warning. Often symptoms can be very vague and diagnostics don't always catch everything soon enough or provide enough to go on that would make anyone suspect things much worse are on the horizon. Doctors are not gods and tests are not perfect and we don't know everything about why some pregnancies develop complications. I'm sure that's not the answer you want to hear, but it is absolutely the truth. I work in this environment everyday and see how ambiguous pregnancy complications sometimes can be.
I wish you the best of luck in sorting this out and working through your grief.
To add: I also want to mention about predatory lawyers. They are trying to make money, and they know that there is money to be made, especially if a case goes before a jury - people are sympathetic to families who lose children and feel like blame has to be assigned, and to who else than the care providers.