Orders not completing properly with PayPal Standard

I'm not sure. When I was testing things I had PayPal Express installed, but when ever I tried to go through the checkout procedure I got errors. I relayed them to my hosts who told me the server my site is on is too old to update, but they offered a "free migration" to what I suspect is going to be a much more expensive service.

I'm still talking back-n-forth with them about it, because I already pay a huge monthly bill to them, but experience all kinds of downtime and problems like this one. I might be forced to look for new hosting, but that remains to be seen. I've been with the same company for ten years, so hopefully we can work something out.

Sounds like they can and should move you to a different server.
If you're using v1.5.5 as you initially suggested, then you should be compatible with a newer server.