Based on your description of the facts, since the tenants vacated the unit after you served them the 3 day notice to pay or quit, an unlawful detainer action (or eviction proceeding) is unnecessary. The purpose of an unlawful detainer proceeding is to gain lawful possession of the premises. Since the tenant gave you the keys and it appears you have other facts which indicate that they have surrendered possession, you now have legal possession of the premises. You should concentrate on re-letting the premises. You may bring a small claims action for breach of contract against the Tenant who may have prematurely vacated the Premises for back rent.
However, your situation is very fact sensitive and the landord/tenant laws are quite tricky. You should hire an attorney to help you.
Furthermore, this answer should not be considered legal advice and does not create an attorney/client relationship.
No. You got what you wanted --and they left the unit in pristine condition-- since they complied with the notice to pay or quit by quitting, so you can't sue them for unlawful detainer when they're not unlawfully detaining your unit. If you did sue them, they'd win and they'd be the prevailing party and they'd recover their attorney's fees from you. Why you'd want to hurt these tenants is unclear and disturbing.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
No, you absolutely CANNOT proceed with an eviction action. If the tenant is not in possession when you file your lawsuit, the tenant is not "guilty" of unlawful detainer. The tenant is not "detaining" your property. The tenant complied with your notice.
The fact that you want to "hit the tenant's credit" is spiteful and, if you go through with the unlawful detainer action, illegal. You may be setting yourself up for lawsuit if you do.