I just got a Dark Brotherhood follower, and he does not have the best armor or weapon... Can I give him better armor, a better weapon, and if he dies will he respawn?

Only through a glitch. Try reading the section in the wiki. http://www.uesp.net/wiki/Oblivion:Followers#Equipping_Followers And to answer the latter...Yes.

How do I get an item to weigh 0, and if they die with a sword I gave them, will they lose my sword?

You can not make a item weigh 0, but certain things do, mostly clothing but also bound items. If they die with your sword, you can simply pick it up like anyother time someone dies and drops a weapon.

I try to give my follower shrouded armor, but it keeps saying ''You cannot place items in a container while pick pocketing.''

Use the blue highlighted links in the Equipping Followers page I gave you. I don't want to go into how to details in this forum. Those links will tell you how to obtain the weapons and armor you need.

Good luck

That's because it has weight. Read the section on reverse pick pocketing.