Can LTM redirect traffic behind GTM?

You can log the requests in LTM:

when HTTP_REQUEST {
  log local0. "Client IP is [IP::client_addr]"
}

This information will be in the /var/log/ltm logfile.

It would be nice if there was a field on the HTTP profile stats screen that showed you how often the fallback host was utilized, but I'm not aware of a way to view this. You would need to take a packet capture and analyze the HTTP response to your clients to see if any an redirects to your CDN are occuring.

GTM can log requests with an iRule or by setting a db variable.

Rule:

when LB_SELECTED {
   log local0. "LDNS [IP::client_addr] assigned to [LB::server addr] for request [DNS::
rrname]"
}

DB Variable:

At the CLI, type b db GTM.QueryLogging enable

Picks in GTM are the number of times the GTM pool member is picked for resolution.

If you want one virtual server on an LTM to get more picks than another, you would need to set a ratio method instead of round robin.

Regarding monitoring, yes it's possible for one protocol to work and not another. Firewalls, ACL's, etc could be in play here. The user manual is pretty complete with the instructions on setting monitoring. Personally, I use the bigip monitor and let the LTM tell the GTM what is and is not up and functioning. However, there may be a reason for you to validate path information and therefore an additional monitor may make sense.