Do dividend and rental income count towards eligible earned income for Roth IRA contribution limits?

Dividends are not earned income. The rental income is unearned unless it is earned income from self-employment, this would be someone who is in the business of renting property, otherwise, it is passive income.

Thank you! Just so that I'm clear, that means that my dividends and rental income DO NOT count towards the $135,000 IRA income limits correct?

No, the contribution limit is based on Modified Adjusted Gross Income which INCLUDES dividends and rental property income.