Fix docs preview slider collapse, auto-sync Mintlify cache

[jlowin]

Sliders rendered in docs <ComponentPreview> blocks were collapsing to a 12px thumb with no track. The cause turned out to be the preview shadow-DOM wrapper in embed.tsx, which uses display: flex; justify-content: center; align-items: center to center the preview inside its card. That makes the prefab root Div a flex item sized to max-content, so anything inside the cascade using width: 100% (sliders, progress bars, anything that wants to fill) ends up resolving against a content-sized parent and collapses. The playground doesn't hit this because it mounts into a normal block element with max-width.

Fix is one rule: [data-prefab-mount] > .pf-app-root { width: 100%; }. The flex wrapper still handles vertical centering, but horizontal sizing now cascades like normal block layout. While I was in custom-handlers.mdx, I also modernized the examples to the idiomatic PrefabApp context-manager pattern with js_pipes / js_actions defined as dicts above:

js_pipes = {
    "stars": """(value) => '★'.repeat(Math.floor(value))""",
}

with PrefabApp(js_pipes=js_pipes) as app:
    with Column(gap=2):
        with Row(css_class="items-center gap-4"):
            Slider(name="rating", value=3, min=0, max=5, step=0.5)
            Text("{{ rating | stars }}", bold=True)

Verifying the fix surfaced a separate annoyance: Mintlify's dev server copies docs/ static assets into ~/.mintlify/mint/apps/client/public/ once at startup and serves from the cache. Edits to renderer chunks or playground.html aren't visible until the dev server restarts because Mintlify only watches MDX. The second commit teaches build-docs and build-playground to mirror the affected assets straight into that cache, so renderer/playground iterations show up on the next page reload — no more manual restart. It's a no-op when Mintlify hasn't been run locally, and the build helpers are extracted into a new build_helpers module to keep cli.py under its line limit.