Yes, as soon as a transaction is received by a node (assuming that node is configured normally) it will broadcast that transaction to its peers immediately following validation. Of course, if the transaction has already been mined (included in a block) it will fail validation. This is necessary because unless that one node has a lot of hash power, it will be very difficult to mine since it would be the only one to know about the transaction. There would also be a lot of conflicts if the mempool wasn't shared.
On your second question:
When you submit a transaction, the wallet software should deduct that from your available balance (assuming it was not rejected). In Bitcoin (and other related currencies), if the transaction is not confirmed (included in a block) within a certain timeframe (Bitcoin is 2 weeks), then it will be removed from the mempool, and the coins are free to spend again. Otherwise, you can do a replace by fee to override that transaction and send coins back to yourself for a small fee.