Did I forget important things to visit?

Hi, I'm Adrian and i travel to Japan by myself arriving in May 1 at 22:00. Starting from May 2 in Tokyo, i have 15 Full days to travel until May 16 (Plane take off around 22:00 Tokyo).
I bought a 14 days JR Pass which ill activate on May 3 so i can use it until last day.
Itinerary is :
May 2 : Tokyo
May 3,4,5,6 Kyoto(festivals in this dates)
May 7 Osaka
May 8 Nara
May 9 Hiroshima + Miyajima
May 10 Himeji + Iga Ueno + Nagoya(i could skip Iga Ueno if is no time)
May 11 Gifu (Takayama + Shirakawa-Go)
May 12 Fuji (lakes area) + Hakone
May 13 Nikko
May 14,15,16 Tokyo
Do i forget important things to visit? Should i cut from some of these and add some more days to others? Do all of them are worth visit? Mount Koya?
Thank you!

I don't think you'd be able to see Iga-Ueno and Himeji in the one day, I'd suggest skipping it. Then you could go from Himeji straight to Takayama, to give yourself more sightseeing time the next morning.
You also won't be able to see both Fuji five lakes and Hakone in one day coming from Takayama, its a very long trip. I strongly suggest doing some research on train travel times by using Hyperdia: http://www.hyperdia.com/ and work out what you want to see while you're there, so that you know you have enough time in places, and you're not just running around from one place to the next.

I'd suggest basing yourself in Kyoto from May 3-8 and just doing day trips to Osaka and Nara to save time. moving from one hotel to the next. Same with Tokyo: base yourself there from the 12th to the 16th, and do day trips to Nikko, Hakone and the five lakes region from there.

Days 9-12 are just impossible. Remove half.

As @Hakata14 says this is really far far too rushed. The suggestion from @Mon is a good one, if you can then try to reduce your hotels to Tokyo & Osaka/Kyoto & do day trips that would be better. However I am concerned that this trip starts in A WEEK from now and you are just asking this question.

What festivals in Kyoto are you talking?

I don't think there will be a big festival in Kyoto these days,
though there will be hold the small scale festivals like Fujinomori matsuri.
http://souda-kyoto.jp/travel/saijiki/05_sub.html#Event_0000247

Just to add, the Fuji Five Lakes and Hakone are on opposite sides of the great big mountain of Fuji. So they're not destinations you'd typically want to do in one trip. And in May, it is risky to expect a view of Mt. Fuji from Hakone. (If it were in colder seasons, you can get a great view though.)

Meanwhile, there are direct buses from Takayama to the Fuji Five Lakes area such as Kawaguchiko (Lake Kawaguchi) if you're interested.
http://bus-en.fujikyu.co.jp/highway/detail/id/38/#timetable

I respect your itineraries. But for myself,I will buy greater Tokyo pass=7200 yen and explore whole 14Days in kanto region.I think 14 Days is not sufficient. So I will return Japan next time to explore other areas.

I could stay 1 day less in Kyoto so i have 1 more day for the rest! You re right about no big festivals, but there is small shows as horseback archery on may 3, geisha dance, traditional horse racing on may 5 and another with traditional stuns done on runinng horses on may 5. Ill do as you said, nights in big citys like Tokyo or Kyoto from where i start my day trips! Thank you!

I know is only a week, but ill try to visit how much i can for each day, if not than its ok! I can skip the 5 lakes area and do only Hakone, but i have to at least get closer to mount Fuji for some good photos