I lived in Tokyo for 3 years, so I think I can help out.
I'm glad to see you're planning to spend 2 days at an onsen. Spending a night at an onsen was the best thing about my trip when I first visited. I've never been to hakone myself (I usually went to Gunma prefecture for onsens), but it seems like you and some other posters are on top of this.
I went to a Japanese baseball game one time with a couple of friends, but they're the ones who planned everything so I can't give you too many details. It was fun though, and I think there were even fireworks.
You say you think that the location of the Newotani hotel in Tokyo isn't great, but that location is fantastic! City center of Tokyo and close to a few train stations. You can reach anywhere in Tokyo's center within 30 minutes from there. Most places will only take you ~15 min to reach.
I'm not an expert on nightlife, so I can't help you all that much in that regard. (unless you like karaoke, then definitely check out rainbow karaoke in Shibuya. That place is the nuts in terms of karaoke).
I am however an expert on food in Tokyo. First of all let me say that there's no way you can go to Tokyo for 4 days and eat everything, so you're going to have to make choices on what you want to eat the most. Here's a few of my recommendations:
Tonkatsu
https://tabelog.com/en/tokyo/A1305/A130503/13114695/
Hands down #1 in the world. They don't take reservations, so you'll have to wait in line for what used to be about 20-30 minutes if you showed up early. The best Tonkatsu in the world will cost you about 20-30$.
Unagi
https://tabelog.com/en/tokyo/A1305/A130503/13029734/
In my opinion the best Unagi in Tokyo and probably the world. You have to make a reservation, but the shop owner does not speak English. Even if your hotel makes a reservation, it may be hard to eat here, because the old man who runs the place speaks no English. ~20-40$ depending on how much you eat.
https://tabelog.com/en/tokyo/A1324/A132401/13003509/
If somebody argued this is their favorite Unagi in the world, I couldn't blame them. It's also top notch. I had to wait in line for 30 minutes, but according to the tabelog link you can make reservations. They get quite a few tourist visitors I think, so eating here should not be a problem. ~25-50$ depending on size.
Sushi
https://tabelog.com/en/tokyo/A1301/A130101/13156961/
Best Sushi I've had outside of Sukiyabashi Jiro. It will set you back about 100-200$. There's a ton of other michelin star sushi to choose from, but this is my recommendation. This one has 2 michelin stars.
https://tabelog.com/en/tokyo/A1305/A130501/13077980/
It's kinda hard to find in the basement among shops and metros. However, I'm convinced this is the best value lunch in the world. I lived nearby and must've been here ~200 times in 3 years. 5$ for a lunch set. I'd be impressed if you can spend more than 20$ and finish your food. Despite being dirt cheap this is top 3 most delicious sushi I've had under 50$.
Oyakodon
https://tabelog.com/en/tokyo/A1309/A130905/13034307/
I took my siblings here, and it ended up being the best meal they had in Japan. They still talk to me about this place constantly. I recommend you go for lunch. It will cost ~10$.
Yakiniku
https://tabelog.com/en/tokyo/A1321/A132102/13008910/
Top 3 Yakiniku I've had. We couldn't even finish our food and it ended up only costing ~30$ per person. It's a little bit away from the city center but worth the trip.
Fine Dining
https://tabelog.com/en/tokyo/A1321/A132102/13008910/
http://www.ukai.co.jp/english/yokohama/
3rd best meal of my life, and best piece of food I've ever had. It's Teppan Yaki, in a gorgeous mansion. Highly recommend. Costs ~150-200$.
https://tabelog.com/en/tokyo/A1306/A130603/13046855/
Kaiseki with 2 michelin stars. Really fun and delicious meal. Best kaiseki I've personally been to. I believe it cost ~150$.
Izakaya
https://tabelog.com/en/tokyo/A1303/A130303/13163239/
For me possibly the best decorated restaurant I've ever been. Lunch is great and cheap for ~10$. I've had both my best Sashimi and best rice ever here for lunch. Didn't think I'd ever say I love rice, but the rice I ate here I loved. Dinner will be a bit more expensive starting at about 30$.
I could write much more about food in Tokyo, but it may be better to ask further questions if you have any.