Where you are in your planning right now sort of strikes me like someone who has never been to Hawaii saying "I want to go to Waikiki Beach and also go to a luau. Oh, and maybe ride a mule on Molokai. Is there anything else I should do there?"
The good news is you have lots of time to do some research. This site (Japan Guide) has plenty of good information about Tokyo (and Hakone and Mt. Fuji). You don't have very many days there, so you are going to want to find things that are of interest to you personally. Other people's recommendations are going to be the things they like (or the things they think other people should like), and they might really not be your cup of tea. So look to see what else you can do in the area besides the things you already mentioned, and then learn a little more about them.
Another site I recommend for Tokyo information is this one:
https://www.timeout.com/tokyoYou have a lot of time before you can even consider booking a trip. Keep going back and looking at different pages there and making notes of things that look really interesting to you. Japan Guide is great for the "classic" tourist things, but Time Out Tokyo is a little more dynamic. (But make sure things that tickle your fancy will actually be available at the time you are there.) There are plenty of other information sources as well. And you can keep coming back to the forum and asking questions.
Have you really researched what you will find at any given fifth station of Mt. Fuji? It takes a lot of time to get there, the transit isn't particularly cheap, and you might end up being disappointed if what you really want is just a great view of the mountain and possibly a nice little day hike. (I believe there are better places for both.)
From Tokyo, Hakone is a nice day trip (if you can get up early--I wouldn't recommend it if you like to sleep in and not leave the city until late morning) or overnight stay. An advantage of doing it as a day trip is you can (hopefully) pick a day with good visibility. In that case, if what you are mainly interested in with Mt. Fuji is a great view, then you can get it there. But it will be a fairly fast-paced day, especially if you try to do the full round course. Hakone is also frequently recommended as an overnight stay, in particular to experience an onsen ryokan. But in that case, you will want to book in advance, so you could possibly get skunked on the weather/visibility. (Dry weather does not in itself mean good visibility.) Another consideration about when to go to Hakone is to try to avoid the weekend if at all possible. In any case, do a lot of research beforehand. There are many ways to experience Hakone, and if you play it right it can be fabulous but if you don't use your time effectively it can end up being disappointing. As for the onsen ryokan experience, it is a high point of a Japan trip for many people, but you should know what you are getting into. (You have to take your clothes off to get into the baths, you will probably sleep on a tatami mat, you might get served fish at breakfast, etc.)
An alternative to Hakone if you are interested in an excursion with Mt. Fuji views is the Fuji Five Lakes area. This might be a better choice if you like roller coasters, for example. There are other excursions with Fuji views as well, but Hakone and Fuji Five Lakes are the ones that are most famous and have the most readily available tourist information.
Transportation in the Tokyo metropolitan area can be quite complicated but you can simplify things greatly by getting a prepaid IC card (look into Suica). Don't try to save money with day passes (such as a three-day metro pass or all-day JR lines pass). Those things are great if you really know what you are doing. (Wait until your tenth trip or so.) Otherwise, an IC card is the only way to go. Learn what it does and doesn't cover, and it will be your best friend. For Hakone transportation, there's a good chance the Hakone Free Pass is what you will want. Read up on it.