feat(web-ui): move side questions into an auxiliary panel

[GCWing]

Summary

• move BTW side questions into a dedicated auxiliary panel so the main thread and side thread can stay visible together
• add session navigation and source-linking improvements across the nav panel, chat header, and right-side canvas for BTW flows
• refine chat input, model selection, tooltip timing, markdown tables, and file operation cards to better support the updated side-session experience

Test plan

• Verify creating a /btw side question opens the side session in the auxiliary panel and keeps the parent session in sync
• Verify switching between main and side targets in the chat input sends messages to the intended session
• Verify session list actions, source back-navigation, and BTW markers focus the correct parent turn
• Verify file operation cards, tooltip behavior, and markdown table rendering still behave correctly in the updated UI