If you booked thru a website, when you clicked checkboxes and/or clicked boxes relating to booking your reservation, you effectively signed for your hotel. Signing for stuff at the hotel is just duplicating what you already agreed to online. Many, many businesses are finally moving away from this needless paperwork.

When you check in, the most one has to do is show a photo ID and credit card. I don't recall doing that much of the time. You've already acknowledged the rate when you made the reservation.

When you check out, same thing. There's nothing to sign. You'll get a receipt after the original checkout time. Since you appeared to agree to leave early but pay for your entire reserved stay, they probably didn't actually do anything to check you out at that time. I would think you'll get an emailed receipt within 24 hours of the original checkout time.

As far as worrying about signing for stuff, most purchases made with a credit card under $25 don't need a signature. When you pay at the pump at gas stations, you don't sign anything. When you submit stuff at work, you probably send things securely, acting as a signature. Physically signing for stuff is old-school.