(1) What kind of railway pass allows me to take Odakyu Limited Express "Romancecar" from Shinjuku to Hakone?
I don't know. Follow the link I posted before.
(2) What JR Pass should I buy for going from Hakone (箱根) to Kamakura (鎌倉) and Enoshima (江之島)?
I don't know. Find it in
JR English page. I don't think JR has discount ticket for ODAWARA to Fujisawa. Normal fare would be around 600 JPY.
(3) Does such JR Pass cover from Kamakura (鎌倉) to Enoshima (江之島) by Enoshima-Dentetsu (江之島電鉄)?
I don't think so. Find it in JR pages for sure.
(4) What other places will such JR Pass cover?
I don't know. Find it in JR pages.