The alternative is the Okakyu trains and buses to gotemba station/IC (the Okakyu shopping tour bus departs too early (8:30 am) for me). Besides the more expensive 'Romance Car' express to Gotemba, is there any timetable for the cheaper regular trains (I've found the schedule for the buses)? Also, I cannot find the fare information for both the odakyu buses and trains.

Yes, all seats are reserved. However, if the bus is not fully booked, it should be possible to purchase tickets and seat reservations just before departure on the same day.

Would it be full easily?

Considering that it is a weekend, yes, I think there is a considerable risk that it might be fully booked.

If yes, is it possible to make phone reservation while I don't speak much Japanese?

They do list a phone number on the following English page, so you can at least try it:
http://www.jrbuskanto.co.jp/bus_route_e/cotimep01.cfm?pa=2&pb=1&pc=j00...

Odakyu buses from Shinjuku to Gotemba IC cost 1540 yen one way.

No.

Also, it seems that for most trains, kids under 6 are free of charge. Would it be the same for these buses (provided they don't occupy any seat)?