1.

You don't have to introduce me to Sally. She's an old friend of me.

You don't have to introduce me to Sally. She's an old friend of mine.

2.

Let me introduce me. My name's Tim Tracy. Your husband and I went to school with ourselves.

Let me introduce myself. My name's Tim Tracy. Your husband and I went to school with each other  /  one another.

3.

Everybody in my class do their Latin homework on their own.

Everybody in my class does their Latin homework on their own.
(Even though everybody is singular, in modern English we usually use their when referring to an indefinite pronoun).

4.

The teacher has ten dictionaries, so there are enough for the students to borrow each one.

The teacher has ten dictionaries, so there are enough for the students to borrow one each.

5.

It was Monica by herself who worked out the meaning of the coded message.

It was Monica herself who worked out the meaning of the coded message.