At arrival the process is Immigration (regular line), then luggage pickup and then you might have a customs check (you don't have to fill out a form unless you want to declare something). The arrivals area is on the first floor. You might have a prearranged transfer or otherwise it's probably best to get an official taxi (Taxi Green, Taxi 365 or Directo -- they are located right outside of the terminal building). You might get caught up in the morning rush hour if it's a weekday.

For departure:
I personally wouldn't stay at the airport hotel unless there was a situation like an overnight connection, it will be much faster to go from Miraflores to the airport very early in the morning that in the afternoon or evening. I would estimate it should take around 35 min at 5am. I usually try to arrive 3h in advance just to have a buffer if there's a problem with traffic or like a big line somewhere in the process, but mostly because I like to relax at the lounge... I don't think this is as critical early in the morning (because it's less likely that you'll have a traffic problem).

The process at the airport is: you'll be dropped off at the third floor, go to check in/luggage drop, then security check (carryon etc). Immediately after you can use the automatic immigration gates (right hand side to the regular line) if you have completed Migracheck (you can do this up to 48h in advance at the website or at the modules available at the airport -- for foreign non-residents this is only available for departures). I haven't seen a big crowd on the regular immigration side in the morning but it could happen I guess.
You'll now be airside, the shops and restaurants are in a central location and they don't assign the departure gate too far in advance. Just in case, some gates can be a bit far.

I hope you have a great trip!