My mother and I spent a night at the food court in the Lima airport in 2005 and while neither of us slept, it wasn't a horrible experience. There were probably 30 or so tourists, many of them Americans, whose international flights arrived around 11pm-midnight and stayed in the food court before early morning flights to Cusco. It was clean and we felt safe because even though the food court was closed (and if starbucks was there at the time, it too closed) because there were workers around all the time. This site will have up-to-date information if you decide to spend the night in the airport. http://www.sleepinginairports.net/

The Ramada wasn't open in 2005 but that could be a good option, although when we went there had been an increase of attacks on taxis and buses leaving the airport carrying tourists. If the room rates are high, Wyndham Rewards points might be a good alternative.

We didn't spend any time in Lima so I can't speak to a daytrip during a layover that but I don't recall any place to store luggage and we couldn't check in/check our bags for our flight until 1 or 2 hours before even though our international and LIM-CUS flights were both on LAN.

I would spend 1 more day in Cusco. You'll be exhausted by the time you arrive in Cusco (luckily most hotels allow you to check in at 8am) and might be affected by altitude sickness.