I have bought weapons in GTA Online, but they are not in the weapon wheel. How to equip them

Instead of pressing -> at the door, walk through the door to the range itself. There should be a glowing blue ring at the far end of the room that triggers the shooting range hobby.

It works just like any other job. Once you trigger the event you can invite people to your lobby (or not, solo works too) before launching the session to start shooting. The challenges are different from story mode though, since they are built as competitive challenges intended for more than one player.

You are only allowed to keep weapons you have purchased in Ammunation. Weapons you have picked up during missions etc are only available until you log off from GTA Online.

You also lose the weapons if you die or change session.