Yes, you take a train to Cerbère or Portbou (some French trains continue to Portbou) and then a Spanish regional train to Barcelona. Spanish railways haven't shared all of their timetables with the rest of the world so these regional trains are missing in many planners. You can find the timetable here. No reservations needed in Spain.
My experience is that interrailing in North-Western Europa is a lot easier than countries in the South-West, like France and Spain.
See answer above.
The answer to this has also been posted here a few dozen times-often in the same Qs when there were complaints about this high price. WIZZair can probably also fly you from Cluj or Iasi to BCN for even less.
By train; Get off the FR TGV latest in Perpignan (you come from Romania, so I guess the route goes via ITaly?-in that case you hardly need a TGV, as these all go to/fro Paris or far north. Pass border IT-FR Ventimiglia-local train to Nice (local free trains in FR are named TER), then to Marseille. There are also fairly fast TER trains-about ev 1 -2 hrs. Avignon (connect from Marseille at tarascon)-via Montpellier to Narbonne-then to Toulouse or direct/or change to Perpignan-Port Bou=border post just over border. From there RENFE broad gauge=means always change run to BCN as Catalunya express-also about ev 1 - 2 hrs and without any extra to pay They make several stops in BCN-not just the Sants main estacion. However-the route from RO-via HU-SLO-IT-FR this way by train is really slow-many, many changes. I guess will probably take 3 traveldays. It may in the end be better to go via Germany-München-Strasbourg.
Coming back: the ES trains run to Cerbere=border post in FR and change there-same route as above. OR if you want to explore more take inland route to La Tor de Carol-=french border, close to Andorra (buslink)-then TER to Toulouse or sometimes overnite train to Paris (seats 10€ on pass)
FREE trains on RENFE are named cercanias or rodalies-in Catalan. Major stops have gates-you have to ask to open them at counter and show pass valid for that day. There are also a very, very few regional express-even to Madrid, but they take 12 hrs to do that