Can one wallet send/receive bitcoins from/to multiple sources simultaneously?
You seem to be missing Unspent Transaction Outputs (UTXO) as a piece of the puzzle here:
Every transaction on the network results in one or more UTXO. You can think of them as a bill with the value of the received transaction. Every time you make a transaction,  the UTXO you put in get destroyed and new ones spendable by the recipient get created. Since each UTXO is unique, they cannot clash with each other (you getting a bill from one person doesn't stop you from getting another from someone else), and you can only spend them once you have received them.