Well, when you die you don't always get your weapons reset, although everyone should lose their loadout after failing a mission in the challenge mode. However, my patch covers that glitch.
The bug with having the teleporters disappearing once you get on the startmap after someone died could be an error with connecting to the worldserver. Basically, the map looks like that, but the teleporters are added by the 'worldserver' (aka coopordie's server).
If you want to be able to keep the teleporters, you have to play on the "Regular mode" , which is located on your (the host's) profile page. (When someone dies, he should lose his weapons but he can still revisit the reached units)
Also, the mod is called "Coop or die" because --theoretically at least-- the players have to cooperate. Otherwise, they all will fail
.If you want to grind your weapons back, unit 2 contains a supper shotgun right in the beginning which is quite useful, as it's one of the most efficient weapons. And no, there isn't an option to keep the weapons or get some in the beginning.