Is union recognition legally required to, say, organise a strike? Do workers who participate in a strike organised by a non-recognised union expose themselves to getting fired?
When strikes are legal is really complicated and depends on industry.
The thing to understand is that labor law in the US was created to "reduce labor strife." Unions existed before they were recognized as legal. What labor law tries to do is make an orderly process with the goal is no strikes and a mutually agreed-on contract.
The company recognizing the union is the first step towards a contract. Basically it's saying "I understand this organization represents these employees and I should negotiate with it."
This reminds me of the German approach to street marches. They are allowed, must be registered with the police, and the police will ensure there are several policemen around for each protestor just in case any of them "try anything". The police form a box around the protest, restricting it from going anywhere not approved and restricting people from seeing the signs; they also blare their sirens so that the chants can't be heard outside the box. The cost of all this is, of course, billed to the protest organiser.
So as you can see, they have incredible free speech! They allow just anyone to carry out a street march!
As someone living in Germany, what you're saying is not true. It is correct that protests have to be registered, that's the case for pretty much all developed countries. It is also correct that the organizers are responsible to cover costs. But the police doesn't restrict people from seeing signs, doesn't blare their sirens. They mostly stay on the side while handling the traffic, and intervene if the protest becomes chaotic. In the vast majority of protests they do literally nothing other than being present.
Compared to the US where multiple protesters have been killed on video by ICE this year Germany is a dream of free speech
The thing to understand is that labor law in the US was created to "reduce labor strife." Unions existed before they were recognized as legal. What labor law tries to do is make an orderly process with the goal is no strikes and a mutually agreed-on contract.
The company recognizing the union is the first step towards a contract. Basically it's saying "I understand this organization represents these employees and I should negotiate with it."