Grammatically yes, "I heard" is the correct form, however most native speakers in England will use "I hear" I'm not exactly sure why we do it but if you are speaking to a native you can use either form and it will be correct in speaking.

"I heard that you are..." is the most correct from a strict grammatical point of you. However, "I hear that you are..." is perfectly natural still because is the colloquial version, this version is also the most commonly used because it has the nuance that you curious about the information you heard, rather than just asking about information in a factual way.