I have been playing for about 4 hours now, as that is all I have been able to play.
In that time, I noticed several examples where I warp into a system with a few other
ships with numerous enemy vessels. We each pick a target, in two instances, I selected
the Negh Var and others picked the Bird's of Prey or raptor class. In each of those cases
I was one or two torpedoes from sending those Klingon souls to Sto vo kor when someone
else fires a shot at the klingon and gets the drop.

What system is the game using to determine who get's that drop?

The group leader gets to choose from a few options:

Need or Greed
Round Robin
Free for all
Master Looter

You were probably running with Round Robin...

How can I set it so that I am solo in all future missions? No team.

In the social window, there's an option that lets you set such things... setting to no open grouping will disable the auto-grouping. If you then also change the option below that to allow invites, you'll have full control in all grouping situations.

Thanks for this info!

Yeah he had to be in Round Robin. Most of the time when I auto join a group or do secure the sector quests it seams to take a few loot times before I get my chance.