Do you have any specific destinations in Europe in mind? Europe is a big place and different airline alliances have different amounts of coverage. (Although it is usually easy to fill in gaps with purchased tickets from low cost carriers.) This might make it easier to recommend a preferred alliance, although I would say that American is probably your best bet right now due to the large sign up bonuses on credit cards.

You WILL have to be a little crazy to make this happen in the amount of time...but that's what Flyertalk is for. And there still are no guarantees that there will be seats open when you are wanting to go. You say June or July...do you have any flexibility to go a little earlier, like May, or later, like September? Those are less peak times.

Also, while this isn't what you asked, you may want to consider paying for the flights and accruing hotel points to use at nicer hotels. The flight itself is a short part of your trip. Someone already mentioned SPG which is a nice option. Check out the Priority Club forum too, sometimes nice hotels go on the "pointbreaks" (discounted award) list, although you probably can't book those til much closer.