[Feature]: Add a system tray or menu-bar icon for background status and controls

[ElliotDrel]

Before submitting

• I searched existing issues and did not find a duplicate.
• I am describing a concrete problem or use case, not just a vague idea.

Area

apps/desktop, desktop lifecycle, and T3 Connect

Problem or use case

When T3 Code is running, there is no persistent system-level indicator comparable to the tray icons or menu-bar icons used by other background applications.

That makes it difficult to tell whether T3 Code, its backend, and the local T3 Connect environment are still running after the main window is closed or hidden. It also makes background startup less understandable because the user has no obvious place to confirm status or reopen the app.

Proposed solution

Add a persistent T3 Code icon in the Windows system tray, macOS menu bar/status area, and the appropriate Linux notification area where supported.

The icon should make it obvious that T3 Code is running and provide quick access to basic status and lifecycle controls.

Expected behavior

• Show a tray/menu-bar icon while T3 Code is running in the background.
• Clearly indicate whether the local T3 backend or T3 Connect environment is online, starting, disconnected, or experiencing an error.
• Provide a menu with at least:
  • Open T3 Code
  • Show/Hide window
  • current environment or connection status
  • Quit T3 Code completely
• Optionally show useful lightweight information such as active/running session count.
• Make the distinction between closing the window and quitting the application clear.
• Offer a setting for whether closing the main window keeps T3 Code running in the tray or exits completely.
• Preserve active sessions and remote availability when the window is hidden but the application remains running.

Why this matters

T3 Code can act as a persistent remote environment through T3 Connect, so users need a dependable way to know whether it is still available. A tray/menu-bar presence makes its background behavior visible and reduces accidental shutdowns.

It also pairs naturally with a Launch at login option: T3 Code could start quietly in the background while still giving the user an obvious status indicator and a way to open or quit it.

Smallest useful scope

A first version could include:

• a persistent tray/menu-bar icon;
• Open/Show and Quit actions;
• a simple online/running status;
• a setting to keep running when the main window closes.

More detailed session status, notifications, and per-environment controls could be added later.

Related request

See #5510 for launching T3 Code automatically at system login.