There are protocols to distributively create a secret, either a key or a random number. Searching for Distributed Key Generation (DKG) should help.
Depending on what you want to do with the secret determines if you can keep it secret. There are distributed signature schemes. I am not aware of how to do distributed decryption1 or distributed symmetric encryption.
This depends on your secret sharing model. If you require all nodes to participate in any action, then no amount of adversary-controlled nodes will give the adversary the secret. If you require less than all the nodes to participate, say $t$ nodes, then once the adversary has $t$ nodes, he gets access to the secret.
As mentioned in 2., you can choose a threshold, $t$, of nodes that you need to participate.
As mentioned by @yacovm in a comment, Shamir's Secret Sharing is a common threshold secret sharing scheme.
1Not strictly true. I have seen interesting ways to do distributed decryption. Say if the encrypted information is audio, you have multiple audio files such that listening to any one of them doesn't provide any information, but if you set up the speakers correctly and stand in the right spot, you can hear the original message. Of course, this is not general purpose and comes with many caveats.