Q:

Rest API and UUID

One of the reasons, and probably the main one, to use UUID is to avoid having a "centralized" point responsible for creating and assigning ids to resources. 
That means that, for REST APIs, the clients could (and should) be able to generate, and give the UUID for a certain resource when they POST that specific resource for the first time. That would minimize problems related with successfully posting a resource for the first time but not getting the ID back as response (connectivity problems for example). That can result in a new post for some of the clients, generating duplicated resources. 
My question are: 

Having UUID generated by clients and being exposed in the REST API is a best practice? 
Are there any other alternatives to that?

A:

REST does not really care if the UUID is generated by the server or by the client. It just needs a unique resource-identifier in form of an URI.
What form the URI has, is not important to clients and servers - only that they are unique and may be obtained by clients (HATEOAS). You need of course also a resource on the server side which is able to create the sub-resource for you and understands that you want to provide the UUID instead of generating an own one. Instead of a UUID you could f.e. also use a url-encoded title of a blog-post or like this question a combination of hash-value and question-title 31584303/rest-api-and-uuid to uniquely identify a resource.
Generating a UUID by a client is in my opinion not used that ofen in practice, but I may be wrong on this matter. The actual question is rather, why should a client really provide an own UUID instead of letting the server create one? The client is, IMO, only interested in getting the data to the server and having some way to retrieve it at some later point, which will be provided through the location header returned in the response of a POST request. 
If connection issues are an actual concern, you could let the client send an empty POST to create a resource and send back the location within the header. The content is than added via a PUT request. 
There still can be some connection issues involved:

request does not reach the server
response does not reach the client

While the primer one is no issue for the client as well as for the server as no operation is executed and the request can simply be resent, the latter one will actually create a resource at the server side, though the link never reaches the client. Therefore the actual resource is in an unreferencable state unless you provide a way to iterate over all resources, which contains also the empty ones.
A server can have a cleanup thread in the back which removes empty resources after a given amount of time. If a client sends a further empty POST request but this time also receives the URI of the created resource, he can update the state of the resource via PUT. PUT is idempotent. If the server did not receive the request, the client can simply resent it. PUT has the semantic of updating existing or creating a new resource if it is not yet available. So, the server can create the resource in that case with the provided content. If the request did reach the server, a further update does not change the state of the resource.

