I was arrested for a DUI but the police officer never read me my rights. Will the case be dismissed?

Usually, the answer is no. Police are only required to read you Miranda warnings if you are placed into custody and then questioned.

Do I have to do roadside maneuvers like touching my nose or walking a line?

No you don't. Those roadside maneuvers as they are called by police are voluntary. You have every right to decline to participate.

Do I have to take a blood or breath test?

No you don't. However, if you refuse to take a chemical test your license is likely to be revoked for 1 year. If you take the test and your result is too high your license is likely to be revoked for 90 days on first offenses.