Hi!
I'm going to do my best to answer as many questions as I can.
1) Define what you mean by see it properly? Do you mean stand on the 5th Station? See it from closer than Tokyo? The classic place to see it from is Kawaguchiko also known as Fuji Five Lakes. People do go as a day trip, but it's a long day. It is generally recommended to stay overnight, though lodging options can be pricey.
2) If you get a National pass, it is good from whatever day you have it started until it's been either 7, 14, or 21 days. So if you start it on a Monday, it's good for 7 days or until Sunday. So if you're not using it for 1 week of your 21 days, it's time to really think about if it is a good value. There are flexible passes but those are only for JR East. A National pass might not be your best option. The pass is a great deal if you are traveling in such a way that it is a great deal.
People on a budget also use buses or even fly. My friend who lived in Osaka tended to fly to Tokyo because it was cheaper than the bus and much faster and as a resident she couldn't get a national rail pass.
3) Some people rent SIMs. I don't know where you are from but I'm in the USA and I just use my regular cell phone service plan with Project Fi. Yes, I use roaming data, but the additional I pay even with roaming data and calling the US is still less than the cost of a rental SIM or Pocket WiFi. I'd ask whatever plan you are on about it to see what the cost will be.
4) Booking for the Alps depends on where you are going and where you hope to stay. The Alps are pretty large. If you're talking Takayama during the Takayama Festival or Matsumoto while the sakura is in bloom or many places during Golden Week, well yes, pre-booking can be best and prices can REALLY vary. Takayama during the Takayama Festival is much pricier than Takayama on another date.
5) Accommodations in the Alps vary by where and when, much like accommodations anywhere. In general, camping if you don't need to rent equipment is very cheap, mountain lodging can be 8000 yen a night with food, there are also luxury hotels in ryokans in the mountains that will cost you a lot more. It all varies.
6) Visiting beaches to swim or to look at the water and admire the scenery? I have seen on the occasional warm day in early May people on the beach playing in the sand, but the water is still going to be awful cold unless you are one of those people who really like 12-13 C water. The beaches will not have lifeguards or anything else set up for use. Beaches outside of Okinawa don't tend to officially open until July. The Japanese will think you are crazy if you go in the water, but people do walk around the beaches and take pictures.
7) No you can't do Mitake-san and Hakone in one day. They're not very close to one another. Depending on how early you can get up in the morning and what you wanted to do in each you can do Hakone as a day trip from Tokyo. I had a friend who once did that, we just wanted to visit Yunessun and not do anything else there. So in the early evening she went back to Tokyo and I went onto Kyoto. I've done the circuit several times so had no need to do it again. I've not been to Mitake-san, so can't speak for whether it is possible to leave in the morning, hike around and then go back to Tokyo but I would think it is doable since it takes only 100 minutes from Tokyo Station to Mitake Station.
8) I didn't think there were any volcanoes in Hokkaido currently erupting... if you mean dormant with a recent eruption history there is always Asama-san which is quite close to Tokyo. I believe Sakurajima near Kagoshima is the most active volcano of the 4 main islands. Kyushu also has a few other relatively active volcanoes like Aso-san and Unzen-san. But Honshu is just full of dormant volcanoes outside of Fuji-san, as in most mountains in Japan are dormant volcanoes. Azuma-san not far from Fukushima is currently under watch because of concern for volcanic activity. Please see here:
https://www.japan-guide.com/e/e7754.html Shirane-san near Kusatsu onsen and Manza onsen is also currently under watch:
https://www.japan-guide.com/e/e7406.html Heck, in 2016, Owakudani was partially closed and many hiking trails are still not open because of volcanic activity.
https://www.japan-guide.com/e/e5203.htmlSo nope, you don't need to go to Hokkaido for volcanoes. You can, if you want, but if volcanoes are the goal you don't have to travel anywhere near as far.
9) I find a lot of hotels seem to have Universal outlets and I've not had a problem with either my computer or cell phone with my US plugs. I've no idea about British ones. I've not needed to bring a converted in decades.
10) What is the 440,000 yen covering? Does that include airfare to get to Japan? Again, it sort of depends on your plans are and what you want to do.
11) Many mountain huts need to be reserved in advance or they might fill up. Just to be clear, many places in the Alps will really just be opening for the Season in mid April. Depending on where you want to climb there will still be snow, like as in a lot of it. I found some pictures for someone else who was asking about climbing near Kamikochi in May and the whole top of the mountain was just COVERED in snow. So do be aware because people have died climbing in Japan when they've not been prepared. Higher altitudes in Japan can be very slow to lose their snow. I share this story often and I know it is anecdotal, but when driving on a recently opened road near Towadako in late April I got caught in a blizzard with barely any visibility. Places that are just opening in April get a lot of snow.
Many Mountain Huts have ways to reserve on-line or via email. Just search for the names. I booked a cabin in Kamikochi via email. I've had hotels help me book some activities but I've never had them book other accommodations for me, that sounds a little weird.
Unlike @Guest I do tend to actually use Booking.com. I've not yet had any problems and I've booked 3 different trips to Japan with them. I've found most of the hotels I've booked have been cheaper via Booking when I compared with the actual website. I like Booking because they tend to include all taxes in their prices so you know what the price is ahead of time. Depending on where you are going their selection is more limited. I'm looking at some less touristed places so I am unsure what I will use for booking, though I am also trying to figure out how I want to end my trip. (Visiting Nikko or visiting someplace else.)
