First, the options are not just car vs. Eurail pass. You can also buy individual "point to point" tickets in advance online; these will be non-refundable and non-exchangeable, but in exchange for these restrictions, the tickets are MUCH cheaper. And for Cinque Terre to Barcelona, you can consider flying from Pisa Airport if you won't have a car (it's a long train ride).
Second, yes, if you have a Eurail pass, that only covers the "base" fare. You have to buy any seat reservations on top. For domestic trains in the Netherlands, Germany, and Switzerland, reservations are almost always optional (some exceptions exist). But for the faster trains in Italy, France, and Spain, as well as most international trains, reservations are mandatory. Furthermore, France severely restricts the number of pass-holder reservations. Particularly for a family of 4, you would need to book the France train reservations at the time you buy the pass; if you're doing that, you might as well buy discount advance tickets. When you buy a point-to-point ticket on a high speed train (whether advance or last minute, discount or full fare), the reservation fees are automatically included.
For a very good discussion of these issues, see The Man In Seat 61's post, Should You Buy A Railpass?, here: http://tinyurl.com/bkw4u6c
To find out where to look for the advance tickets on each route to see the prices, go here: http://tinyurl.com/bo8x6o6. Note that Rick's maps of train routes and costs do not reflect the advance purchase, non-refundable prices, so do not rely on them; go directly to the source, which is usually the website for the country where the train originates (again, the link in this paragraph will give you specifics). Also when looking, be sure to put in the number and ages of all travelers; there's often a discount for two or more traveling together, and in some countries your children may qualify for lower fares, depending on their ages.
To find flights within Europe, use Skyscanner: http://www.skyscanner.com/