chore: rebrand project to GopherGate

Updated all naming from LLM Proxy to GopherGate. Implemented new CSS-based branding and updated Go module/binary naming.

cmd/gophergate/main.go

@@ -5,9 +5,9 @@ import (
 	"log"
 	"os"
 
-	"llm-proxy/internal/config"
-	"llm-proxy/internal/db"
-	"llm-proxy/internal/server"
+	"gophergate/internal/config"
+	"gophergate/internal/db"
+	"gophergate/internal/server"
 
 	"github.com/joho/godotenv"
 	"golang.org/x/crypto/bcrypt"
@@ -48,7 +48,7 @@ func main() {
 	s := server.NewServer(cfg, database)
 
 	// Run server
-	log.Printf("Starting LLM Proxy on %s:%d", cfg.Server.Host, cfg.Server.Port)
+	log.Printf("Starting GopherGate on %s:%d", cfg.Server.Host, cfg.Server.Port)
 	if err := s.Run(); err != nil {
 		log.Fatalf("Server failed: %v", err)
 	}