Paypal supports TLS 1.2 The older protocols are being retired.
TLS 1.2 Upgrade
The most secure protocol for sharing information on the web today is Transport Layer Security (TLS) version 1.2. PayPal is enabling support for TLS 1.2 for all secure connections and in 2016 will start requiring its use. You will need to verify that your environment supports TLS 1.2 and if necessary make appropriate updates. PayPal is updating its services to require TLS v1.2 for all HTTPS connections in June of 2017. After that time, all TLS v1.0 and TLS v1.1 API connections will be refused.
If you are not using TLS 1.2 you will receive that error.
Looking at the error you received
[03-Feb-2017 06:36:47 UTC] cURL error:  Unsupported SSL protocol version
it looks like the PHP CURL version used on your server is out of date and does not support TLS 1.2.