Nowadays, in formal English, it’s correct to use the same pronoun case in a compound as you would use by itself. In your examples, we need the nominative case (subject). We say “I have known…” so we say “Tom and I have known….” We say “He gave it to me,” so we also say “he gave it to Tom and me.” This is how it was always done in Old English, and also conforms to Latin grammar, which was considered important in English education not too long ago. This is how I speak. That being said, the Norman French conquest had a huge effect on the English language. Although this mostly relates to vocabulary, there is also a strong tendency in the spoken language to use French grammar in compound subjects — the emphatic pronouns or “pronoms toniques.” These days, this is considered colloquial. As an example, people may say "me and Tom know..." “Tom and me know...” just like “Tom et moi” — regardless of case (subject or object). “Hypercorrection:” Some people, who have been corrected when they used “Tom and me” as the subject, use the form “Tom and I” all the time — even as an object. This is simply wrong. Example “He gave the money to Tom and I.” Horrible.

The best way to say them are: Tom and I have known each other. She and Tom have known each other.

You need the subject pronouns. Which are 'I' and 'she.' She and Tom... Tom and I... I don't know why, it's some kind of collocation but 'I and Tom' and 'Tom and she...' both sound awful and no one would say them. If you want to complicate the issue a little, some people do use the object pronouns here, particularly in speech (me and Tom/Tom and me, her and Tom) but it's not considered grammatically correct.

Hello, Tom and I have known each other Tom and her have known each other