14 days in Japan - Itenerary help

I know there is plenty of ppl asking for itenerary help, so any advice will be greatly appreciated. My 2 friends and I will be going to flying into Tokyo on June 21 and leaving July 4th. Here is our 'vague' itinerary:
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day 1- arrive at 3pm. spend day/night in tokyo
day 2 - activate JR rail pas; take 5hr train to hiroshima. stay overnight there.
day 3 - head to osaka - 1½ hours. stay overnight there.
day 4 - spend day/night in osaka
day 5 - head to kyoto (¥2930, 14 minutes). stay overnight there.
day 6 - head to nara and back. stay overnight in kyoto.
day 7 - spend day/night in kyoto
day 8 - JR rail pass expires. head back to tokyo (¥13, 520, two hours 22 minutes). stay overnight there.
day 9 - 11 Visit Nikko,hakone for sure. and possibly Kamakura, Yokohama, and Mt Fuji
day 12 - spend day/night in tokyo
day 13 - spend day/night in tokyo
day 14 - board plane at 7pm
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we simply trying to figure out our high level view so we can start booking hostels. i have a few random questions:
1. should we spend more time at hiroshima rather than juts a day?
2. is two days in osaka too much?
3. is three days for kyoto/nara enough?
4. are nikko and hakone worth staying overnight at or just simply day trips? will the rainy sky reduce view of mt. fuji?
5. Also, which city would ppl recommend to see hot springs?

Much appreciate for anybody that has answers to my questions or overall opinions.
thanks

should we spend more time at hiroshima rather than juts a day?

The peace park can be covered in half a day. Miyajima also requires about half a day at a minimum. So, if you do not mind a fast travel pace, 24 hours are enough. Of course, if you want to see more than just the main sights and/or travel at a slow pace, then you should add some time.

is two days in osaka too much?

It depends on the person. Some people skip Osaka altogether. But for others, one day is not enough. What do you want to see?

is three days for kyoto/nara enough?

For people who like a fast travel pace and have a not a particularly big interest in traditional Japan, three days are enough. But for those with a big interest in traditional Japan or those who like to travel at a slow pace, I recommend to add some time in Kyoto.

are nikko and hakone worth staying overnight at or just simply day trips?

If you want to see only the shrines and temples of Nikko, a one day trip is sufficient. But if you also like to explore the national park, an overnight stay is recommended, especially if you like to travel at a slow pace.

For Hakone, I always recommend to stay overnight, because one of Hakone's main attractions are hot springs, which are best enjoyed by staying overnight at a ryokan with hot spring baths.

will the rainy sky reduce view of mt. fuji?

On rainy days and even fair days but bad visibility, Mount Fuji will not be visible from Hakone. I myself have been to Hakone about ten times, and only on three occasions I was able to see Mount Fuji. And only once were the skies were perfectly clear.

Also, which city would ppl recommend to see hot springs?

Hakone.

If you want to cover both Hiroshima/A bomb sites and Miyajima, I'd add an extra day. From Tokyo, you'll probably be arriving in Hiroshima around noon, and it'll take the rest of the day to do EITHER the Peace park area OR Miyajima. I'd split them on 2 different half day trips, so you can see the shrine and get some views from Mt Misen on Miyajima.

I'd skip Osaka or take it as a day trip from Kyoto, though you'll have plenty to do in Kyoto for those 3 days.

Generally speaking, Osaka's fun is around shopping and eating, and you can probably do those same things in Tokyo on even a grander scale. And the 'sights' of Osaka aren't must-sees, so if you're on a tight schedule, I'd give it a miss.

As for the Fuji area, I agree with Uji totally. Go for some relaxation and onsen soaking--if your luck is good, you'll get to see Fuji unobstructed, but even without the 'view', you can unwind and sightsee the area. Totally worth it after rushing about for a week or so.

... of which 3 days were in Kyoto.
I really wish I would have given Kyoto some more time. It's just such a lovely city with so much to see and do (not only temples!). If you have any interest at all in traditional Japanese culture and architecture, I would recommend adding a day or two in Kyoto.