Sending BTC from old QT Client V0.7 - Will coins arrive?
There haven't been any Bitcoin hard forks in the last 3 years so your old client should still work fine. Of course it won't be able to use new features like SegWit but that shouldn't matter.

Does your v0.7.2 client have the "Transactions" menu option. Double clicking the pending transaction will tell you it's status. If you don't have that menu option where are you getting the transaction ID from?

If you're still getting nowhere with your old client then in your shoes I'd take the following steps:


If you haven't already backed up your wallet then in your v0.7.2 client do File->Backup Wallet and save it to a USB drive or somewhere safe,
Download the latest Bitcoin Core client and install it,
Run the new client and it should update your wallet if it needs to,
If your blockchain is not synchronised then unfortunately you may have to wait a long time (days) for it to catch up,
Try another test transaction with the new client.


There are other options that you could do with your existing client, such as extract your private key with the Debug Window->Console, but that could go wrong very easily. Your number one concern is obviously not to lose your private key(s) and even if it takes a number of days to sync your new client that's a small price to pay.