How to Perform an SEO Audit on an Existing WordPress Site? Where to Start?

I have an old WordPress site that's been running for several years. It has a lot of content, but search traffic just won't pick up. I want to perform a full SEO health check on the entire site. Seeking help from the experts—here are my questions:

1. What structural elements require thorough inspection (URLs, categories, tags, internal links, etc.)?

2. What common pitfalls at the technical level (such as duplicate pages, pagination, archive pages, and internal search results) need to be blocked or standardized?

3. Are there any recommended plugins or online tools that can help quickly identify issues?

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