IF the person who most likely will take me to a small claims court knows that I'm currently illegaly in the country can she do something about reporting me? Would the small claim court arrest me?

Small claims court wouldn't arrest you. If immigration was notified and cared to arrest you, they might meet you there since its easier to arrest someone if you know where and when they might be there.

Not likely to have any immigration enforcement at small claims court, unless there are allegations of criminal conduct or security issues.