How HTTP/2 affects the implementation of a proxy server? Especially, for example, when a client sends an HTTP/2 request to a content server that only supports HTTP/1.x, should the proxy server tranform the HTTP/2 request to HTTP/1.x request before directing the client request to the content server? And upon receiving response from the content server, should the proxy server transform the response to the HTTP/2 format before sending it back to the client?
Yes, it is exactly as you say. Conversion from HTTP/2 to HTTP/1.1 must happen in one direction, and from HTTP/1.1 to HTTP/2 must happen in the other case.
In practice this means that although HTTP/2 the protocol doesn't need a traditional text-based parser, a comprehensive HTTP/2 server needs an HTTP/1.1 parser, not only to work with clients which are HTTP/1.1 only (among them crawlers) but also for talking to inner applications.
By use, one of the most important application protocols is FastCGI. FastCGI also requires parsing of HTTP/1.1 responses from the application and conversion to HTTP/2 responses to the client.