Traveling to Europe (Schengen Agreement)

Ireland and the UK are not in Schengen, so we have:

Belgium/Netherlands (1½ weeks)
Northern Germany (2½ weeks)
Poland/Czech Republic/Slovakia/Hungary/Austria (3 weeks)
Back to Germany (2 weeks)
France (3 weeks)
Fly out of Paris, France.

1½ + 2½ + 3 + 2 + 3 = 12 weeks = 84 days, You are legal!
But, you need to list out the actual days and add them up, the arrival and departure days are both included in the count.

BTW, it is 90 days in 180. If you (for example) put the UK in the middle, the clock stops when you leave the Schengen Area. But, when you return, it restarts again from where it left off!

As the first poster noted, you should be fine. You have 90 days in the Schengen countries, that includes any partial days (If you arrive from the UK at 10:00 pm, those two hours are still a day, both in the Schengen zone and the UK). Don't worry about trips back to the US, unless you work the 180 days, it does not help. For the UK and Ireland you get 6 months so you could do more time there, Croatia is sort of in limbo, you should get time outside of the Schengen zone, but they are half in half out, the other Balkan states are outside of Schengen, as is Turkey, but that is further afield. Technically, between the UK and Schengen, you could be over there indefinitely.

The other option would be to look into a visa. It is a process, you have to show financial support, health care, some other things, You will need to pick a resident country and seek a visa there, but that does not prevent you from travelling. As a retiree, it is a bit easier, but for what you want, just managing the schengen restrictions may be enough.