1. yup, 7 days is just right for Tokyo. u can make 1 or 2 side trips to nearby places such as hakone and yokohama too
2. good to stay near stations along the JR Yamanote line coz tt is the main line u will be using to get ard Tokyo. popular places to stay will be Shinjuku, ueno, shinagawa.
japan is cramping down on airbnb. not sure if there is going to be problem in the future.
3. train. cab is way too expensive in japan. by HSR u mean their bullet train (aka shinkansen)? shinkansen is mainly for travelling long distance out of the city and it is expensive.
if u really want to take shinkansen, u can consider getting JR Tokyo wide pass and using if for shinkansen to utsunomiya (for nikko). and use if for other side trips such as to hakone.
http://www.japan-guide.com/e/e2361_06.html
however, this is not the most cost effective way coz there are other passes that covers some of these places (e.g. hakone free pass, tobu nikko free pass), just that they do not include shinkansen. so getting this pass is more because u want to experience shinkansen