Learn Turkish: Ben elenmedim, sen elendin! - I wasn't eliminated, you were eliminated!

You're right that the endings -m and -n already tell us who the subject is. However, in Turkish, the spot right before the verb is the focus position (vurgu). By explicitly placing the pronouns ben and sen in this pre-verbal spot, the speaker is heavily emphasizing the contrast: I wasn't eliminated, YOU were!

What is the root of the verb elenmedim and how is it formed?

The root is elemek, which means to eliminate (or to sift). To make it a passive verb (to be eliminated), we add the passive suffix -n because the root ends in a vowel. This gives us elenmek. Then we simply add the negative -me, the past tense -di, and the personal ending -m.

If I wanted to add a time word like dün (yesterday), where would it go to keep the emphasis on the pronouns?

To keep the emphasis on ben (I), it must stay in the focus position directly before the verb. You would put the time word at the very beginning of the sentence: Dün ben elenmedim. If you instead said Ben dün elenmedim, the pre-verbal focus would shift to dün, changing the meaning to: I wasn't eliminated yesterday (but maybe I was today).