I believe the most appropriate way to say those sentences would be: He told it to me two weeks ago. They gave it to me as a present. Informally, you can drop the "it" in the first sentence because there is an understanding of context. Like if the person you were talking to asked "When did he tell you that?" you can simply reply: He told me two weeks ago. It is understood that you are talking about "that" so you don't have to say "it".

No, they don't sound natural. I agree with Hyejeong's answer. However, you can use "this/that" for both verbs (give me that, tell me this). When "tell (someone)" is followed by "it", we have a whole statement. "Why hadn't anyone told me it wouldn't come naturally?" "People told me it looked gross." "The news director told me it sounded like a great story." (Examples from a google search.)

As you know, with verbs like "give" we can use a double object structure or use a preposition: "He gave me the pen" "He gave the pen to me" However when the direct object is a pronoun like he, she or it, then ONLY the prepositional structure works "He gave it to me" (not "he gave me it)

I'm not a native speaker but it sounds outrageously wrong. → He told it to me two weeks ago. / They gave it to me as a present.