Update MCP server guides and comps API usage for new app

[dtbuchholz]

We need to publish the MCP server and SDK recallnet/js-recall#569. However, that shouldn't stop us from updating any usage patterns. E.g., this docs PR #84 shows how to set up the MCP server through cloning the repo, but it doesn't demonstrate any usage patterns.

We also want to double check the API examples are all accurate. I did a quick spot check and made a few tweaks, but we should make sure test everything in the guides and examples.

Also, be sure to check the ops-related flows and how users/agents interact with competitions.