- Do the active connections to the Espers network influentiate the staking? The most I've ever had is 1 active connection, always.
No the number of active connections does not influence your staking probability, although 1 connection is a bit low If your only connection drops while it's your turn, you will miss the stake. Take a look at the wallet syncing guide in my signature.
How can I sign a transaction and how do I acquire the private key?
All of that happens automatically, you have nothing to do. You can read how the original PPCoin PoS system works here: https://bitcoinmagazine.com/articles/what-proof-of-stake-is-and-why-it-matters-1377531463/ Your wallet will do all of these things on its own if online.
Every block could be your block so you don't want to miss it.