A bitcoin node is a user which is connected to the network through a client. This node can perform transactions, try to build a block inside a mining pool, or just exist in the network. Are there any further actions for a node?

Nodes also relay blocks and transactions to other nodes. They also validate incoming blocks and transactions and keep track of the current, longest valid blockchain. That's pretty much it.

If a node exists inside the network how can he verify the transactions?

There are a set of mathematical rules that determine if transactions are valid. The nodes check the transactions against those rules. If they meet the rules, they are valid.

Furthermore is there a paper or something that explains things in a lower level of how exactly the system works in case of transactions and mining? I do not want a description but something more technical.

Almost every question you can think of is answered, in technical detail, somewhere in the Bitcoin wiki.