The flashing ghost icon just means that there is a scannable object nearby, justifying the fact that walking away from an area makes the scannable object farther away and unscannable. The "Scannable objects" are ghosts that, using your ghost, you can interact with to see further insight into the lore of the game. No other benefit is received for finding these objects. Check here for more information.
You described the differences pretty well.
In Destiny 1, shaders were normal items - they were not consumable. There was only one shader slot, and it applied to all your armor (weapons were not affected (related, most exotic weapons had one or two ornaments that worked similar to shaders, but were consumable)). You could change the shader at will, and the shader would not be destroyed. Later on in Destiny 1's development they added a shader kiosk where you could get a new copy of any shader you have ever found which would allow you to effectively use the same shader across all your characters.
In Destiny 2, shaders are now applied per item and can be applied to weapons, sparrows, ships, and ghosts in addition to armor. Shaders are also consumable - once you've applied a shader, you can't use it again. If you replace it with a different shader, the first shader is gone. When a shader drops, you typically get a few copies of it (I've gotten 3 at once, not sure if that is guaranteed).
The main issue some members of the community have with the new system is that in addition to being consumable (which means you tend to think about applying a shader, since you can't use it again), is that you can buy them with real money via the Eververse store. The conspiracy theory is that shaders were made consumable so that Bungie would make more money on microtransactions.
Best Answer
Gear will be saved the moment you acquire it.
Mission progress will be lost if you quit. Most of the time, missions are short enough that it should be considered "acceptable" if you have to quit early. Once a mission is complete and you return to the solar system map, that mission counts as completed.
The intro tutorial is actually a collection of missions, and I'm pretty sure if you quit during one, it will bring you back to the most appropriate starting point.
There are checkpoints in some missions that it reverts to if you die, but these aren't stored after quitting.