Flowers are the plant's way to do sex, as you know from school days---if you leave the ovaries (where the seeds develop, pods behind the flower in lilies) on the plant, the plant will think of nothing except making the next generation & will devote all its energy to ripening those seeds.

Cut off the pod & the plant will stil think of nothing except making the next generation, except now it will have to concentrate on making the flowers for next year, when it will try again.

When the stalk loses its green, cut it down to a little above the ground. This is not likely to happen until well into autumn.

Giving it some more well rotted compost won't hurt, but don't do it when the plant is about to go dormant for winter.

I think it is an LA I saw in the Scheepers catalog---go check out the online version.

Beautiful boy, too!

When it finishes blooming, cut off ONLY the part of the stem from where the flowers start - up. Leave the leaves to let the lily replenish the bulb with the nutrients it creates from summer sun.

Not sure what zone you are in -- if you get good hard freezing in the winter, it should be in the ground, at least six inches deep. If it's now in a pot, you can just tip it out and into a good hole, and firm the soil up around it.
If you don't get frozen, you could probably leave it in a pot, but it probably would be happier in the ground.