Bullets do not go through Deathtrap. However, rockets and other things that explode on contact (glob gun shots for example) do go through deathtrap.
How I tested it:
- I cleared an area of enemies and summoned Deathtrap.
- I then positioned him between myself and a corrosion barrel. Close enough that even if I missed him due to Anarchy I couldn't hit the barrel.
- I then unloaded several clips into him, aiming at the barrel on the other side of him.
- The barrel didn't go off.
- To double check that I wasn't mistaken in my aim; I waited for him to disappear and then shot once, blowing up the barrel.
Additionally, to try to rule out ricochets off of Deathtrap I performed another test:
- I have 5 points in Close Enough (50% chance to ricochet)
- I summoned DeathTrap and positioned him in front of me, with the barrel off to my side.
- I unloaded several clips into him, aiming at different points and strafing to ensure that if I was ricocheting that it shouldn't have been able to miss.
I noticed that no ricochet trails were appearing on screen, and the barrel did not explode.
Playing Gaige in a "no aiming" style requires a different mindset from the other Vault Hunters. The Ordered Chaos tree works better when using a weapon with a low Magazine Size but high Reload Rate. Generally, that means you're shifting away from automatic weapons such as SMGs and Assault Rifles that often have Magazine Sizes of 20-30+.
One other thing you'll need to force yourself to forget is reloading in between fights. Anarchy stacks based on emptying your magazine or killing an enemy, so the fewer shots you have your current magazine, the sooner you'll accumulate stacks of Anarchy.
Anarchy itself is an odd mechanic. Because it reduces Accuracy per stack, you'll actually want to use a weapon with a decent amount of Accuracy otherwise you'll be doing little damage even at close range (a la Sledge's Shotgun from the first Borderlands). Combined with the need for low Magazine Size and high Reload Rate, your best shot for a go-to weapon is anything from the pistol family. Be a revolver or a repeater, these guns have respectable Accuracy and Fire Rate, without having the relatively high Magazine Sizes you want to avoid.
With this mind, you can stack Anarchy as quickly as possible for use with your other skills. While this isn't so much a "no aiming" style, Gaige's Ordered Chaos tree puts a lower emphasis on aiming than other playstyles. What's making this hard to swallow isn't so much the skill build but the mindset involved. A lot of it run counters to what you learned in other FPSs.
P.S. If you really want tips for skill builds, I would recommend getting the Close Enough skill in the Best Friends Forever tree as soon as you invest a couple of points in the Ordered Chaos tree. With your decreased Accuracy from Anarchy stacks, the ricochet will definitely help. After more points in the Ordered Chaos tree, I would recommend coming back to the Best Friends Forever tree for The Better Half skill. As you want to empty your magazines as quickly as possible, any source of increased Fire Rate helps. In fact, The Better Half skill might actually make high Magazine Size weapons such as Assault Rifles, Machineguns, and SMGs more palatable to use.
Best Answer
The only way that I've found to skip the animation is to go into a menu. So inventory, Badass Rank menu, Mission Logs and so on.
I have not tested this in multiplayer, so it might not work there. Should it work in multiplayer you usually can enter and exit it quickly enough so that it shouldn't put you at too much risk of dying.