So let's say the aircraft can do 5600NM and the route is 5000 but it says that the type of plane can only carry x amount of passengers, is that valid?
The range of an aircraft is shown with zero wind. So depending on your route flown, range might me lower (or higher if you fly with the wind).
Runway length also has an impact on range.
So the passengers it shows that it could carry is the actual number and not the number when I lease my aircraft?
almost. But all aircraft have different weight, engine,fuel tank and winglet options. So when in "route planning" the figure for pax and cargo carried is based on the shortest range aircraft of that type in your fleet. It's only when the route is assigned to an actual aircraft and flown for the first time that you'll see the real numbers. This is why you'll look at EGKK-RCTP for example and see different seats flown each day, for example 250,250,320,250,250,210,210 as the airline has different options flying the 7 day schedule.
Simon