Who's the first person that Odysseus kills and why does he kill him first; after this first man is killed. . . what plea bargain does Eurymachus...

The first person that Odysseus kills is Antinous.  We are not told exactly why Odysseus kills him first.  Presumably, it is because he is one of the two leaders of the suitors and Odysseus wants him out of the way.

After Odysseus kills Antinous, Eurymachus tries to make a deal with him.  Eurymachus tells him that the suitors will make amends for what they have done.  They will pay Odysseus back for all of the food and drink that they have consumed.  They will also each pay a fine to Odysseus.  Odysseus, of course, refuses.