When does the giant door behind the bounty tracker open

It is possible to run over enemies with the Sparrow, but it is difficult. Here is a video of it being done in a Crucible match:

Note: This functionality is no limited to the Crucible. I've run over enemies while patrolling as well as during missions.

For your first question, as I am unsure of which class you are playing as this is only a guess. I believe if you are playing as the titan (striker subclass) you could be experiencing the effects of either the Shoulder charge or Juggernaut perks. After sprinting for a set distance you will see particle effects surrounding your character.

For your second question, these are most likely going to be damage indicators. These will show you the direction that you are receiving damage from.

And finally for your last question, no, your gun will not heal your team mates. this is simply a visual indication of whether the guardian you are aiming at is from your team (green icon) or the enemy team (red icon).