One of your servers is not loading. What’s the first thing to double-check when having a problem installing ESXi?

1- First you have to check your Servers hardware regardless of which operating system you are trying to install in case of the server is not loading at all. 

2- If the server is loading the boot menu but you can't boot to the operating system you are trying to install so you have to check the boot settings (Legacy and UEFI) and to check the boot order, and for ESXi you have to check if Virtualization is enabled 

3- If the installation is booting but you can't proceed a full installation then you have to check for hardware compatibility in vmware.com to make sure if they fit for this installation or not (Harware model and minimum requirements)

4- If everything is supported you have to make sure that you have at least one NIC to complete the installation 

make sure that everything is good and in order and you will be fine.

Server compatibility and server connectivity from console 

First thing to double check is the physical LAN connection to the physical hardware, Log into the EXSi admin console and test connectivity to the gateway, In case of Oracle based HW open admin console by ILOM console. Tip: Knowledge of your network architecture is essential, often issue is with the network. 

The first part of the question is misleading or not clear.

For the second part:  for any issues coming  during esxi installation, verify first that the server hardware you are using is in the Vmware Hardware compatibility list.

The issues that may occur like purple screen or mostly nics missing cases.

Please check the Vmware Comapatability Matrix and proceed

The first thing is to check the server compatibility issue, i.e the HW requirements in VM compatibility guide.