1. Which one will be broadcasted first in the blockchain?
as soon as you send the transaction it will be broadcast and then it takes something around 10 seconds to propagate throughout the whole bitcoin network.
although there are some conditions, for example if you are sending a non-standard transaction nodes can reject it, if you pay a very low fee (like 0) nodes can reject it,...
2. Does timing of the transaction submission plays a role?
not really.
although it may play a role in a different way. for example the mempool may be empty now (less than 2000 tx) and sending a transaction with a low fee (lets say 1 satoshi/byte) can get you a confirmation in next block (10 min on average). but in a couple of hours mempool can grow big for whatever reason (bigger than 50000 tx) so paying the same fee can mean nodes rejecting your transaction or having a confirmation way too late than the next block.
3. Does fee plays a role if everything remains the same? (both senders send 1 btc to the same address at the same time).
yes.
but remember that fees are paid based on the size of the transaction in bytes not amount.
