Do you need a certain apartment to do a heist in GTA Online

Most of the early game missions are triggered by phone calls from Simeon, Lamar, and one of Lamar's friends. These should start up at about level 4. As you level more options in free mode will appear.

I personally had problems with these events triggering automatically on day 1. Day 2 seemed to be better in that regard. Though be weary, there still seems to be issues with the servers not remembering everything you've accomplished.

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.