Appeal deadline questions after an unemployment denial (update)

My claim was denied after 26 weeks of waiting, and the letter says I have a short window to appeal. This is my first time dealing with any of this. The letter lists a deadline and a hearing process, but it is hard to follow.

Thanks for writing this out. I went through something close to it 3 months ago, and it settled down once I kept a simple log of every call and form.