Regarding remote work between Europe and the USA, is there an easy path for US based companies to hire a US citizen or an EU citizen to work remotely from an EU residence?
The US company I work for is remote first and we hire all around the world. Not sure how helpful this is, but our method is that if we have a registered company in their country they will be hired as a typical employee. Anyone hired in a country we don't have a corporate presence in is hired as a contractor.
A global payroll provider effectively acts as the "local employer" (they have a local entity in each country) and contracts out the employee to your company, so while the employment contract is with the provider, you can still sign all sorts of side-letter contracts like NDAs, IP assignments, stock grants etc without issue.
Pros are that unlike a contractor, the employee is subject to local labour laws and protections. Cons are that the employee is subject to local labour laws and protections (and the employer is subject to local labour taxes, which the payroll provider will pass through to you).
In certain countries with employee-favourable labour laws, we found it easier to attract employees this way, as they may be unwilling to forego the protections and social security / pension payments they would normally get as an employee.
[Edit: but beware of the double edge of course - if you ever want to fire such an employee, you may run up against said strong labour laws.]