I understand that bitcoin operates on the history of transactions and miners processing them and that I can still receive a payment when my address/wallet is offline, but I don’t know if the sending wallet has anything more to do then just signing the transaction and submit it to the bitcoin network.
I’m using Electrum as my wallet and have a semi unstable internet connection, so I’m just making sure that if my connection dies the second after I initiate the transaction it will go through (assuming all else is well).
If your transaction is successfully broadcast, you will be able to look it up on various block explorers such as blocktrail.com, blockchain,info, blockr.io, etc. Once the transaction is broadcast, there is nothing else you need to do in order for your transaction to confirm.
Additionally, when you want to receive Bitcoin, your wallet software does not need to be online either. Once the transaction confirms, it is included in the blockchain and your wallet will look it up when it syncs once it comes back online.