How come "informed (past tense)" was used after "keep"? Shouldn't it be "inform"? or is there any word omitted?
No, "inform" should not replace "informed" here. "Informed" is used as an adjective for the verb "keep". "Informed" as an adjective refers to having knowledge about a situation. You can think of this phrase as "keep me knowledgeable about what is happening". "Inform" can be a verb or adjective. Ex: My sister informed me that she's getting married. (Informed used as a verb) Ex: The police kept the reporters informed about the robbery. (Informed used as an adjective)