Help me resolve few queries about bitcoin network ( regarding validation, mining and state of transactions...)
A1. It remains in the mempool of those nodes until a miner decides to include it in the block it's attempting to mine. Making a transaction from node A to node B is a little confusing, maybe you meant from address A to address B?

A2. Yes, nodes that share the same consensus rules (have not hard forked) will have the same blockchain (public ledger). The blockchain determines history of transactions, and the order in which they occurred.

A3. Nodes will validate transactions before they enter their mempool. Miners typically will just mine the block presented by the node (with tx's chosen to maximize profitability in terms of BTC fee/kB). The transaction is not entered into the blockchain (ledger) until a miner finds a valid solution and submits that back to a node. The node then begins to propagate the new block through the network.

A4. Yes, but this does not mean that transaction has been mined in a block yet, it will just be in each node's respective mempool.