Does a pizza lose flavor/texture from cooling down and then heating up again versus keeping it warm from the original oven until eating?
Flavour

Keeping the pizza warm for an extended period will result in some volatile compounds evaporating, thus slightly changing the flavour of the pizza. On the other hand, reheating the pizza (which will probably require the oven to be at a higher temperature than the target pizza temperature) will do the same. It is difficult to predict which one will have a bigger effect, but I expect neither method to stand out.

Texture

This is going to be a bigger issue. Keeping the pizza warm in an enclosed space for an extended period will result in condensation, and make the pizza more soggy than it originally was. Conversely, reheating the pizza in a hot(ish) oven will add an extra dose of evaporation on top of the original cooking, resulting in a pizza and toppings that are dryer than it was when fresh.

Keeping the pizza warm, rather than reheating, might also do some weird things to any cheese present, which might get rubbery rather than melty.

Food safety

Keeping the pizza warm for an extended period might land you in trouble if the temperature is below 60 °C. See this answer and references therein for more information.

How to have hot pizza long after cooking?

This is going to be opinion-based, but I do have some ideas. Ideally, eat the pizza as soon as possible after its initial cooking. Keeping the pizza warm for a short amount of time (say up to 30 minutes) is acceptable. Any longer than that, and I recommend refrigerating the pizza, then reheating it in a dry non-stick skillet over medium heat (cover and/or add a splash of water to steam the top of the pizza if necessary). Also, cold pizza is still pretty tasty.