Use ruma-client for sending requests / receiving responses

It would be nice to push the lower-level parts of converting Rust request / response types to / from whatever the HTTP client library needs to Ruma, which also has an implementation of this that doesn't directly interact with all the Rust SDK's configuration bits and such. With this, there would be a cleaner line between the opinionated API parts in here, and the very basic Matrix protocol interactions in Ruma.

Blocked on https://github.com/ruma/ruma-client/issues/5, and possibly other Ruma improvements.

Start by reading the issue and the linked ruma-client issue 5, then identify the Rust SDK’s request/response conversion code and the corresponding Ruma implementation. The work is done when lower-level conversion and HTTP-client interaction are handled through ruma-client without directly depending on the SDK’s configuration-specific API layer.

Written by the indexing model from the issue text.