Can JAX-RS do Asynchronous Request like JAX-WS?
1) I don't know if the JAX-RS API includes a specific mechanism for asynchronous requests, but this answer could still change based on the client implementation you use.
Can JAX-RS access a web service that is not running on the Java platform, and vice versa?
2) I can't think of any reason it wouldn't be able to.
What does it mean by "REST is particularly useful for limited-profile devices, such as PDAs and mobile phones"?
3) REST based architectures typically will use a lightweight data format, like JSON, to send data back and forth. This is in contrast to JAX-WS which uses XML. I don't see XML by itself so significantly heavier than JSON (which some people may argue), but with JAX-WS it's how much XML is used that ends up making REST with JSON the lighter option.
What does it mean by "JAX-RS do not require XML messages or WSDL service–API definitions?
4) As stated in 3, REST architectures often use JSON to send and receive data. JAX-WS uses XML. It's not that JSON is so significantly smaller than XML by itself. It's mostly that JAX-WS specification includes lots overhead in how it communicates.
On the point about WSDL and API definitions, REST will more frequently use the URI structure and HTTP commands to define the API rather than message types, as is done in the JAX-WS. This means that you don't need to publish a WSDL document so that other users of your service can know how to talk to your service. With REST you will still need to provide some documentation to other users about how the REST service is organized and what data and HTTP commands need to be sent.