You're going to find it increasingly difficult to support your old PHP 5.4 server as time goes along. PHP 5.4 has been obsolete for a couple years already. Out-of-the-box it has no awareness of the modern TLS methods for talking to servers (maybe GoDaddy has patched it somehow, I dunno?)

The "OpenSSL wasn't compiled with SSLv3" errors are simply a reflection that the curltester you're using is trying to use a completely outdated type of SSL/TLS connection. You might try grabbing the latest from https://github.com/zencart/zencart/tree/v157/extras and try again.

CURL error 28 means "operation timed out" ... meaning, in theory, "we tried to connect to PayPal to make a transaction, but the connection didn't complete in the allowed timeframe" ... where the timeframe is typically 6-10 seconds ... which is a LONG time just to connect.

"why" it's timing out is indeed the thing you need to chase down.
It could be your server.
It could be your PHP version.
It could be firewalls.
It could be PayPal is rejecting your incoming transaction requests because they're coming from a format they no longer support. That wouldn't surprise me given you're using ZC v1.5.1 which was released in 2012.
While, yes, I'll absolutely advocate that you should be upgrading, I realize that's not a helpful statement when you just want transactions to work again. It would indeed be helpful to know exactly "why" in order to be certain that upgrading is the key step or not. Upgrading should still be done, but I agree it'd be helpful to get payments working first if possible and then spend time on an upgrade while transactions are still happening.