which is the best loop for tokyo-kyoto-hakone

we land at hanida on 27 april and got our reservations for hotel conrad tokyo from 27- 3rd april. then kyoto from 3 to 6 th may at ishicho at nakagyo . whats the best way to do this loop of tokyo -kyoto - hakone . also we flying back on 9 th from narita airport . whats the best advice ? and should we purchase the JR pass ? how do we save time and money.
also we are vegeterian . will the roykyan serve vegeterian breakfast on request .
thanks

You got to inform the Ryokan prior. Drop them an email. If they don't respond take another place!

Best loop? You've already settled your nights so there is not much else to talk about.

Try to reach Tokyo the day before you leave for the airport.

A 7 day JR Pass is worthwhile. If you can leave Tokyo and return to Tokyo within 7 days great!
If you spend too much time in Hakone, and cannot return to Tokyo within 7 days, you would have to use a 7 day JR Pass and a Hakone Pass. You would naturally save less, but it is still a saving. Furthermore, on holiday it is fine to spend a bit more here and there!

I would recommend
Tokyo
Kyoto
Kyoto to Nara and back
Kyoto to Uji to Osaka and back
Kyoto to Hakone
Hakone round about
Tokyo

Nothing much to plan since Tokyo and Kyoto are fixed

If have additional time, could add in Ise/Nagoya/Kanazawa/Nikko