feat(ui): integrate membership system into existing site

Summary

Connect the membership system (auth, profiles, admin) to the existing US-RSE React app — update navigation, add sign-in/sign-up flows to the UI, and link profiles throughout the site.

Requirements

• Add Sign In / Sign Up buttons to the navbar (replace or complement "Join Us")
• Signed-in state: show user avatar + dropdown (profile, settings, sign out)
• Admin users see "Dashboard" link in their dropdown
• Link board member names on Governance/Board pages to their member profiles
• Add "View your profile" to the Community section
• Update footer with member-related links (Sign In, My Profile, Admin)
• Mobile menu: auth state reflected (sign in vs signed-in user)
• 404 page for invalid profile slugs
• Loading states for auth-dependent UI

Context

Parent issue: #1916
Depends on: #1918 (auth), #1919 (profiles) — this is the final integration step

Implementation Notes

• Nav component needs auth context to conditionally render sign-in vs avatar
• Consider a <UserMenu> dropdown component for signed-in state
• Board page photos can link to /members/:slug once profiles exist
• This is the "glue" issue — it's about connecting pieces, not building new systems

Start with the Nav component and auth context, then trace the Board/Governance pages, Community section, footer, and mobile menu. Review the auth and profiles work in #1918 and #1919 before connecting their existing pieces. Done means each listed auth, profile-linking, invalid-slug, and loading-state requirement works across desktop and mobile UI.