I'd assume the appliance is the best fit for your environment, based on the size of the ESX estate, the limitations of vCSA should not be a constraint, unless you want to use linked mode or an SQL DB. There are other limitations which you can google - vcentre appliance 5.5 limitations. In terms of licencing, having it physical or virtual will mean a Microsoft OS licence, depending on your Microsoft ELA. The appliance will not require an OS a licence.
The vSphere client can be installed on your desktop but you can use the web client anyway if you have flash installed. The web client is frustrating but currently this is where future development is going unfortunately. The C# client will suffice though here. Use the web client as a back up but install the full client on a physical machine as the VM you install it on my be unavailable during an outage. You can install wherever you like, there's no limit to how many clients you install.