UN backdues are paid. They were not paid for a while because of a dispute regarding the structure by which the US was paying a much higher percentage share of dues than it was receiving in benefits. Previously, the US was paying 30% of peacekeeping costs and 25% of regular UN fees. It is now 25% and 22% respectively. This is still not in proportion to benefit but it has been agreed to. The US Congress would not approve paying such a high amount given that the US has only one vote and one veto (20% of veto's) which set in motion the process bywhich the US was not paying it's dues until the matter what settled.
As far as the original question, I don't think the US needs Europe nor do I think Europe needs the US. Both would do well without the the other though with some loss of efficiency (higher costs for some items), and so forth.
I do think however that both are important to each other. European nations give the US important bases for military operations. The US trains many European nation's air force pilots.
Economically, neither can discount the other. I'd say politically that the EU has as much clout as the US economically (tariffs, regulations). The relationships are important.
The UK is the most important ally of the United States. Those nations together can project more power in the world through their navies and in particular their air craft carriers than all other nations combined.
For that matter, Asia too is important and increasing in importance. Japan is by far the world's second largest economy. China is up and coming.
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