"from" is the only way that works for me but I'm British, maybe different in US

Both are correct in real life conversation and no one will ever notice. However, "graduated from xy university" is the grammatically correct form, while "graduated xy university" is incorrect. In writing, especially where it's important that your grammar is accurate, always use "from".

Both statements are fine, though "from" is more common.

In conversation, I would say "I graduated in 2005" as in the UK, it's generally only university that you graduate. I might say "I graduated from university in 2005" but I probably wouldn't use the definite article before "university". In AmE, it might be different.

I live in New Zealand. Here "graduated the university" is incorrect. I graduated in 2005. - fine I graduated from ... - fine