Title: [Feature]: Add support for Claude Code cloud sessions via --cloud flag

[ryannzander]

Description

• Why this matters:
  T3 Code's killer feature is its mobile and remote control surfaces, allowing developers to manage long-running agent threads away from their desks without needing to keep a terminal window open. However, running Claude Code entirely locally requires the user's host machine to stay awake and connected. Anthropic's native claude --cloud workflow addresses this by spawning ephemeral execution VMs directly on their cloud infrastructure. Bringing this capability into T3 Code would allow developers to seamlessly launch persistent, cloud-hosted Claude agent tasks natively from the T3 GUI.

• Smallest useful scope:

1. Cloud Toggle Button: Add an optional "Cloud Session" toggle/checkbox when configuring or starting a new Claude Code thread in the T3 interface.
2. CLI Flag Orchestration: When enabled, T3 Code's process orchestrator should append the --cloud flag to the underlying claude initialization command (e.g., passing claude --cloud "task description").
3. Remote Setup Safeguard: If a user attempts a cloud session but hasn't linked GitHub yet, provide a fallback hint to run claude /web-setup to establish their repo-linking tokens.
4. Handoff Tracking: Detect and listen to the spawned cloud session ID so it displays appropriately within the T3 active thread feed.

• Related but distinct issues:
  None directly mapping to the Claude Code VM handoff, though it behaves closely to the general external sandbox orchestration tracking discussed in [Feature]: On-demand cloud task environments through an external sandbox broker #14793.

• Risks or tradeoffs:

• Cloud sessions are currently limited to a one-way handoff (you can teleport a cloud session down to a local terminal, but you can't push an active local session up). T3 Code will need to prevent users from trying to toggle "Cloud" on a thread that is already running locally.
• Requires the user to have a Claude plan that supports cloud execution seats (Pro, Max, Team, or Enterprise) and an authenticated GitHub connection.