Under the DMCA, counter-notices are not often seen. They give people who are not infringing or are only engaging in fair use a way of saying "prove it or I won't stop doing it".
So, typically a copyright holder will swear out an affidavit under the penalty of perjury that they are the copyright holder and that another work, a site, or a search engine is infringing (or allowing the infringement) of their copyright(s).
Upon receipt of your original notice, the host or presenter of the content has to disable access to the work immediately to avoid liability.
A counter-notice, however, works as a responsive notice saying they are not infringing and that your notice to stop is bogus. While this is important for due process, it also puts copyright holders of very limited materials (who may not have even registered or have much money) at a disadvantage to bigger sites with a lot more money than the holder of a single copyright. It can be abused in that, now, your only recourse is to file a lawsuit seeking a preliminary injunction right away, until the final hearing can be had, where you'd seek a permanent injunction.
So, they will take your work down (or have already) but if you don't file right away, it will go back up when those 10 days expire.
You can go to court yourself, but it is no more advised in this instance than any other (barring small claims which is made for litigants to utilize without counsel). If your copyright has any monetary value, you really need to hire an experienced IP attorney. If it's just something that has value to you, but you really want it down and cannot afford it, you can go yourself and attempt to prove it's yours. If you do this you will need to be prepared with convincing evidence.
When you get to court, you will have the burden of establishing that it is your original work. You must prove that the claims in the DMCA counter-notice are untrue, because they are now claiming that the work is not an infringement: either that they hold the rights to it or their use of it is legal (fair use, licensed, etc.) and that the original notice was filed in error. If you can do that, you will seek damages for the infringement and the abuse of the counter-notice process.