Could you try connectivity of your Arduino device with one of our supported Ethernet or WiFi shields, or connectivity of the ESP8266 as a standalone device using the code from Add New > Device / Widget > MicroControllers > Generic ESP8266 ?
Connectivity using the ESP as a Wifi shield for some other Arduino device is not really officially supported, though I know we have many users doing so using the steps in this post. But because this is an un-official (at the moment) trick, I don’t think its a good place to start troubleshooting potential issues with our Arduino server. In fact it may be that explains why your Arduino/W5100 combo and Wemos → Cayenne connection are working OK while your Arduino/ESP-as-shield combos are not.
Perhaps @kreggly has some ideas for troubleshooting those Arduino / ESP-as-shield combo devices?