External Monitor for Pool

Regarding the Host, that's useful if your web server is expecting a specific hostname so it knows how which web application to route the traffic to (e.g. IIS using host header bindings to differentiate between web applications). if you can directly access the page using the computer name or ip, then the host header isn't necessary (unless you specify HTTP 1.1, which requires a host header). The monitor should check the pool on the port specified on the pool member, so that shouldn't be the issue either. If you try to do

curl -v http://RDSRCH5:52810/CoveoSearchService?versions

This post may provide some more information as well.