>>To ride the Romancecar, a limited express ticket is required in addition to a regular ticket or a Discount Pass.<<
This means that although you can ride the Romance Car (which is a "Ltd Exp" train that calls at only a couple of stations on the way), you would need an additional surplus Ltd Exp ticket for each person. The Pass includes the normal fare (and you can take the Exp which would not require any surplus if you want to, although that takes about 20-30 longer than the Ltd Exp).
From Shinjuku there is one line to Odawara and Hakone-Yumoto operated by Odakyu Railways. They run their regular and express services on the same piece of track. The reserved seating express train (called "Romance Car") requires an additional fee/ticket for the seat reservation. The Romance Car is a little quicker (depends which service you board), and costs and extra 1090 yen each way.
If you get the "Free Pass" that starts/finishes at Shinjuku, then that included the round trip on the regular trains to/from Odawara. If you buy one that commences at Odawara, you can ride (and pay for) any train on any route you choose because there are several options - for example, if you have the JR pass you might take the shinkansen or a JR express train from Tokyo.