Welcome to the forum!
Here's my quick shot:
1. the Oedo Onsen in Odaiba is indeed more of a theme park, but perhaps a good starting point to familiarize yourself with onsen culture. While not really authentic, it's a fun place, there are lots of foreigners too. If you want to see a "real" onsen, you should consider a day trip to Hakone.
2. + 3. There won't be anymore cherry blossoms in Tokyo by mid-April, this year was a very short sakura season due to the low temperatures and heavy rain. You will have to travel north to follow the cherry blossoms. IIRC, they reach the Tohoku region and Hokkaido by the end of April/beginning of May, so it depends on how much time you have at hand.
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This picture (click to enlarge) was taken in Hirosaki, Aomori Prefecture.
We cycled in Golden Week from Tokyo to Hakodate, practically following the cherry blossoms.
Taken on May 3.
4. Isn't the Miyama you refer to close to Kyoto? And what exactly are you looking for?
5. I have never been to Yoshinoyama. Perhaps someone else has more info on that.
6. As for coastal towns that can easily be reached from Tokyo: I recommend you to visit Kamakura and Enoshima (lovely day trip) or if you prefer something more rustic, head down to Miura Peninsula. The town of Misaki at the southern tip is very scenic and offers delicious sea food (tuna, to be precise).