Can one wallet send/receive bitcoins from/to multiple sources simultaneously? Let's say for example I put my public bitcoin address on a website for donations and multiple people for around the world deposit bitcoins simultaneously to that same wallet: 


Will all deposit transactions be valid on the bitcoin network or might some transactions "overwrite" the other?
While receiving bitcoins from someone (not yet a verified transaction on the network), can I withdraw or pay bitcoins using the same wallet given the amount only include previous bitcoins and does include the new deposit?
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.