[Epic 953][Stage I][cloud-api] Make Responses API stateless (no-store only)

[hanakannzashi]

Parent/tracking issue: #934

Implementation: #943

Goal

Retain POST /v1/responses as a stateless, no-store compatibility endpoint without platform-side response or conversation persistence.

Scope

• Default an omitted store field to false and reject store: true.
• Reject state-dependent request fields and flows: conversation, previous_response_id, background: true, file input/search, response-history endpoints, code interpreter, computer, and all server-executed Responses tools. Omitted background and background: false remain valid stateless values.
• Accept only custom client-managed function tools. Cloud returns a function_call; the caller executes it and replays that call plus a matching function_call_output with caller-managed history in a fresh request.
• Reject web_search, web_context_search, remote mcp, MCP approval/list continuation input, and image-generation/editing output models. The separate root POST /mcp endpoint remains independent of Responses.
• Build response and output items in request-scoped memory. Do not create, read, or update rows in responses or response_items, and do not pass a database response ID into CompletionRequest.
• Retire response-history HTTP endpoints: response GET, DELETE, cancel, and input_items.
• Return Cache-Control: no-store across the Responses router, including early rejections.
• Preserve existing completed-response gateway attestation compatibility as described in [Epic 953][Stage I][cloud-api] Restrict stateful APIs and retain stateless Responses #934; it is not response-history persistence.

Acceptance criteria

• Streaming and non-streaming store: false requests complete without persisting response or item content.
• Each successful inference makes one Chat Completions call with no server-side agent loop or tool execution.
• Client-managed custom function replay works without previous_response_id, conversation, or response history.
• Unsupported stateful fields and server-executed tool types return a clear 4xx error.
• Supported inference, billing, rate limiting, and completed-response attestation remain functional.
• Tests cover no-store behavior, client-managed function replay, rejected stateful options, response-history retirement, and the stateless public signature path.

Out of scope

• Deleting or migrating existing data.
• Database schema removal.
• Conversation and File route lifecycle, tracked separately in [Epic 953][Stage I][cloud-api] Retain Files views and deletion for account cleanup #937/[Epic 953][Stage I][cloud-api] Retain Conversation views and deletion for account cleanup #939.
• S3 cleanup.