I agree with others, there is no reason to stay in Osaka. Even for transit connections, a hotel near Kyoto station will do just fine. 5 days in Osaka is pretty crazy, particularly if you are just using it for day trips.
April 1: Leave for Osaka - see Osaka Castle / Umeda
Taking trains in the morning generally means you waste half a day's daylight getting to and from hotels, train stations and on the train. Make long distance journies during the late afternoon early evening.
April 2: Day trip to Nara
Easier to do as a day trip from Kyoto
April 3: Day trip to Hiroshima & Miyajima
April 4: Day trip to Himeji/Kobe - or Okayama/Kurashiki
Much better(cheaper and less time wasted on trains) to spend a night in Hiroshima and hit Himeji/Kurashiki/Kobe as a stop between.
April 5: Leave Osaka to Kyoto - try to see Kinkakuji when we get there
This doesnt make sense. If you are arriving from Osaka, then having to find your new hotel, then having to trek out to Kinkakuji, by the time you get there it will be absolutely mobbed with people. Better to do the most famous sights in Kyoto on days you start in Kyoto
April 10: Leave Kyoto - (i need help!)
to either Kanazawa (2 nights) or Takayama (1 night) - Shirakawago in between stop over
April 13 : Arrive back in Tokyo late afternoon
I would leave on April 9th in the late afternoon for Kanazawa, then head to Takayama.
From Takayama to Tokyo, your best bet is a bus direct or a short bus over the mountains to Matsumoto and then a train. Going all train means a very time consuming and expensive detour.
No point in arriving in Tokyo in the late afternoon, that means you will have wasted 6 hours of the day in transport. Aim to arrive in Tokyo at 9-10pm.
- Should I go straight from Kyoto to Kanazawa or to Kyoto to Takayama first.
Transport to Kanazawa from Kyoto is better.
Just from reading forums, some have advised to go Takayama first because of the scenic ride & that the shinkansen from Kyoto to Tokyo is much newer, or better ride. However, I have heard Takayama to Tokyo may have a view of Mt Fuji? but weather may not permit a view. What's more efficient? Will I miss out on Kanazawa cherry blossoms either way?
The 'newer' train line is the Shinkansen. It goes through industrial areas or is walled off from the countryside so you see nothing.
In any event, you will see the exact ride from Tokyo to Osaka/Kyoto earlier in your trip. Sit on the right side of the train to try and get a view of Fuji, but yes it will probably be covered in clouds.
- Do you think a JR 14 day pass pay off for our travel?
If you insist on making tons of long disntace day trips like you have in Osaka, and take the most expensive train even when there are cheaper/faster options availible then it might 'pay off' but it will mean you will be wasting a lot of your time in transport trying to make it 'pay off.' Not a choice I would make.
You should look into the JR West Kansai Hiroshima pass. It is very useful for getting to and from Hiroshima, and Himeji/Kurashiki and around Kyoto. It is also a lot cheaper.
- Is any part too rushed to do?
As said, staying in Osaka and then wasting 3-7 hours each and every day on the train to do day trips seems silly.
I would add an extra day in Kanazawa/Takayama so it isnt too rushed. Probably at the expense of Osaka.