Well, first off, don't forget that the primary language in Barcelona isn't Spanish, it's Catalan. That said, given the number of Castilian-speakers working in the city, it's a complete mix.

English won't be a problem in any of the places you've listed. In fact, in Amsterdam, the standard of English could put a lot of native speakers to shame. However, it's how you approach things that makes a difference.

A few key words (hello, thankyou, etc) go a long way, as does asking, at the start of each interaction, whether somebody speaks English, rather than assuming it.

As you can read Spanish, you'll do well with the other Latin languages too. That just leaves Dutch, but if you treat that as English with strange spelling and more phlegm, you'll get surprisingly far

Paris is a little different in some ways, and you do hear various stories of English speakers being treated differently to French speakers. However, a lot of the time, it's not the language, it's the behaviour: Paris is often a surprisingly formal city. If you make sure you always greet the person you are dealing with properly, whether in a hotel, restaurant or small shop (a 'Bonjour' or 'Bonsoir' with 'Monsieur/Madame/Mademoiselle') as appropriate, it will go surprisingly far.

Oh, one other tip... If you are still arranging an itinerary, avoid Paris in August if possible. The city empties of its inhabitants for most of the month.