4 days tight schedule

It looks like you are a bit under breaking even with the cost of a regular JR Pass. You might look into a Japanican JR Pass 20% discount ticket.
As for where to stay, close to a station is the most convenient, like Tokyo, Shinagawa, Shin-Osaka etc. There are some cheap hotel chains you might look at, like:

http://www.comfortinn.com/sitemap-japan-JP

http://www.toyoko-inn.com/eng/

http://www.superhoteljapan.com/en/

http://www.route-inn.co.jp/english/

For schedules, see Hyperdia

Sorry, see this link:

Japanican JR Pass 20% discount

If we have 1st class tickets for a JR Pass but they are full, can we use the tickets to get economy class seats?

Certainly, but such an event is very unlikely.