Add server-side search, filtering, sorting and pagination

### Description
Current filtering is suitable while the directory is small, but downloading every project and filtering in Vue won't scale indefinitely.

### Proposed API

```bash
GET /api/projects?q=moderation
&category=Utility
&sort=recent
&page=1
&page_size=24
```

### Acceptance criteria
- query search
- category filtering
- pagination
- deterministic sorting
- total result count
- frontend URL reflects filters

Start by locating the existing project directory endpoint and the Vue filtering implementation, then trace how project results are currently loaded. Add the documented query, category, sort, pagination, and total-count behavior, and ensure the frontend URL reflects those filters; done means all listed acceptance criteria work together.

Written by the indexing model from the issue text.