You’ll need an international license to race in the US but that’s easy to get. In the US, getting an “international licence” simply involves checking an extra box and paying an extra fee when getting your basic annual USA Cycling licence and I’d assume it is at least that easy in Europe.
From the USA cycling rule book:
1H7. Registration
(a) In order to compete in USA Cycling races, riders must present a valid racing license from
USA Cycling or an international license from another national federation that is affiliated with
the UCI.
As for entries, for almost all road races that is done through the USA Cycling website. They have all the races up there, eventually, but often not early enough for real long term travel planning. Search the web directly for races you are interested in to get scheduled dates.