Revert gear to original spot, move Live Room into icon toolbar

[aperson30]

Summary

• The floating "show toolbar" gear is back at its original top-left position and back to a plain icon button - the earlier fix over-corrected by redesigning it into a labeled pill and relocating it, when the actual problem was narrower: it sat directly on top of the axial pane's own "AXIAL" label. Now the axial pane's label (only the axial one - every other pane is untouched) is pushed down below the gear, and only while the gear is actually showing (toolbar hidden).
• Live Room moved from its own labeled pill button near the case ID to the end of the icon toolbar, right after the AI button - same icon-only .vp-tool style as every other button there, with the name appearing on hover via the same tooltip pattern the rest of the toolbar already uses.

Test plan

• npm run build - clean
• vitest run - 298/298 passing
• Manual: hide the toolbar, confirm the gear sits in its original corner and the axial label is visible just below it, not hidden
• Manual: confirm Live Room now appears as the last icon in the toolbar (after AI), showing its name on hover