It's perfectly OK to visit the US on your own, assuming your English and driving skill are reasonably competent. Have been to US many times and never joined a tour. Car rental is cheap. Hotel accommodations are cheap and rooms are large, with New York being the exception. Food portions are generous (aka American size). The only complication is getting a Visa for yourself and your daughter. Let me know where you want to visit (US is big) and I can give you some more advice.
Agree. 大阪 is perfect. You can easily spend one day in 大阪港(天保山) with Lego Playground, Aquarium, Maria Ferry and 天保山摩天輪. (Tips: Get an Osaka Card). And then, 一日去環球影城,一日去奈良餵下鹿. The remaining half day, it will be adult shopping time. That makes a perfect 5 days trip.