Your thinking is right on track.
1 - Yes, and it is recommended to reuse a client instance:
from https://jersey.java.net/documentation/1.18/client-api.html#d4e623:
Client instances are expensive resources. It is recommended a configured instance is reused for the creation of Web resources. The creation of Web resources, the building of requests and receiving of responses are guaranteed to be thread safe. Thus a Client instance and WebResource instances may be shared between multiple threads.
2 - No need, the client itself can handle the requests. In the case of asynchronous requests, it internally uses a thread pool that is configurable.
3 - The Jersey client is thread safe, so threads will not block each other
4 - You can also consider providing the client as a dependency to MyWebserviceClient, and possibly reuse the same client between multiple classes