How is it possibile to verify payments without running a full network node? I've read the Nakamoto's paper "Bitcoin: a peer-to-peer electronic cash system". In paragraph Simplified Payment Verification he said that is possible to verify payments without running a full network node. 

I don't understand how, having a Markle's Tree Root hash, verify payment. In fact, in order to check payment, I need to combine and hash all node of right or left branch, not just the root. 

Moreover, how can choose which Root hash needs to get without see in all nodes?
You are right, in order to prove that a transaction is confirmed in a block you need more than only the Merkle root. The information needed is called the Merkle path and consists of the left/right leaf to your transaction node and the right/left parent, and so on up to the root.

Here is a graph depicting a Merkle path:
http://chimera.labs.oreilly.com/books/1234000001802/ch07.html#merkle_tree_path

The Merkle root and path, and also the block headers of the blockchain, are requested from a selection of random Bitcoin nodes.
Your SPV client cannot verify that the transaction is valid (not a double spend, correct signatures, etc.), but since it can verify that the transaction was included in a block with X confirmations, it can assume that the Bitcoin network accepted the transaction. Thus your SPV node will accept the transaction by trusting that the Bitcoin network only includes valid transactions to the blockchain.