First of all, there's no "euro rail" schedule. ;)
Rather the Eurail Global Pass is valid across Europe on participating railway companies. All major ones are included, don't worry.
There is a timetable change in mid-June and some companies haven't published their timetables after that cutoff date. For example France (SNCF) will do so on 12th March. Italy and Spain likely afterwards. However you may check i.e. April dates* to get a good overview.
July is still far away for the railway world! People rarely book trains months ahead.
For timetables Rail Planner is a good start but I also recommend the DB Navigator app (from the German Railways). It is generally more reliable.
Don't hesitate to ask anything.
Some other advice:
Paris-Rome is a long journey, albeit doable in a day! By leaving Paris at 6am, you're in Rome at 6pm or so. Easy change in Turin or Milan.
Venice - Barcelona is not doable in a day by train. Simply too far away and the railway across the Alps and along the Mediterranean is slow. There's no night train to help. I might suggest spending a night in Nice but it would still be a fairly long journey. With your limited time, book plane tickets.
Last "honest" advice: your itinerary is pretty rushed during peak season. You'll be exhausted, spend hours queuing at every sight, no time to relax, etc.
Personally I'd cut Spain from the trip (or Italy). Instead add a relax 2-3 days in an Italian village/small town, could be near Florence.
*the direct Paris-Milan services restart in April after a landslide-related closure.
