Does running a full node bitcoin core wallet (0.13.1) connected to the network (port 8333 open) with NO send/receive at all, i.e. zero ballance - helps the net or not?
Yes.
If so, how?
It validates the transactions and blocks, enforces the consensus rules, and relays both transactions and blocks to other nodes.
when said that a full node validates transactions and blocks - does it validates any (random) transcations and blocks from the net or just the ones which are send into it as a wallet?
It validates ALL blocks and ALL transactions that are sent by the nodes it is connected to.
what is the difference between a miner and a full-node when it comes to checking (i.e. validating) the thousands of transactions in each block and the history of the blockchain - 24/7, thus securing the bitcoin network?
Nothing.
Miners that don't want to risk being on a bad fork will run a full node and only work with transactions and blocks that the full node has verified.
Full nodes don't trust anything that they receive. They validate every transaction and every block, and only relay the ones that the node determines are "valid".
A solo-miner (or mining pool), is just a full node that is also building valid blocks. The only thing that miners "secure" is the order of the transactions (which came first, second, third, etc). The proof-of-work (hashing) makes it extremely difficult to change that order or to add or remove anything from the historical list.
If a miner tries to broadcast an invalid block, EVERY full node on the system that receives that block will reject it and refuse to relay it to any other nodes.
If a user tries to broadcast an invalid transaction, EVERY full node on the system that receives that transaction will reject it and refuse to relay it to any other nodes.
