I noticed a few days ago in Creation Kit that we have among Skyrims quests several quests which have "ArenaCombat" in their name as well as EastmarchJailArenaFightQuest. I just wonder what's the purpose for them? Are they used for some quests in the game or are they abandoned quests or are they there for a DLC?

They may be implimented/activated for a later DLC, but it's more likely holdovers from when they planned to have an Arena in Skyrim, and then decided against it.

Like Marss said, it's very possible that it's something that will be implemented in the future. The groundwork for future stuff does need to be laid early on afterall.