Sounds like a "gravel bike" would suit you. But I don't think you are going to get anything of decent quality under $500. For sizing I would guess you are looking at a 56 cm or 58 cm frame (medium or large) depending on whether your inseam is short or long for your height. To give you an idea I'm a fraction over 6', with a 34" inseam and 58 cm frames (Large) tend to fit me perfectly. But unfortunately sizing between brands and bike models is notoriously inconsistent. But I think you can safely rule out all small and XL frame at your height. In practice you will probably find that most 56 and 58 cm frames will give you a reasonable fit with some tweaking of stem length.