Replace answer overlay with inline chat input

[yaman]

Story

As a user answering inception phase questions, I want to compose my answers in an inline input field at the bottom of the chat (instead of a popup overlay) so I can see the full conversation history while typing.

Acceptance Criteria

• The current modal "Answer" overlay for inception questions is removed.
• Questions from the AI appear as regular chat messages in the chat stream.
• A persistent input bar at the bottom allows the user to type their answer and press Enter to submit.
• The input bar has a blue left rail matching the user-input chat containers.
• The input bar shows mode/model/phase badges (e.g., "Phase 3 DeepSeek V4 Pro") when an inception question is active.
• Previously submitted answers appear in the chat as user-input containers with the blue rail.
• The input bar is always visible during inception (not just when a question is pending).

Notes

Derived from observations during the inception E2E test (docs/inception-e2e-findings.md, finding #1). The current overlay disrupts context and shows stale options.

[yaman]

Split into smaller stories. See child issues: TUI-007 (#18), TUI-008 (#19), TUI-009 (#21).