GTM and cluster

I will provide you the general idea about that, and I think should answer all the question you posted. If not, let me know.

The HA configuration has no direct relation with GTM. You can have an HA configuration, starting from standalone, active/standby, active/active, N+1, etc.. That is LTM, and other modules that depend on LTM (like ASM).

The GTM/DNS will always be standalone (and implicitly active). If you have more than one GTM, they will work together, splitting work (like monitor), and sharing data (like DNS zone files, if enabled).

When you have LTM and GTM together, is where the confusion starts. The LTM will still work independently from the GTM, but you can configure the GTM to take advantage of the fact that is in an HA pair.

When you configure the GTM listener, that is basically a LTM virtual server, if you use the floating IP, only the GTM in the active unit will process the traffic. If you configure each GTM with a listener using the non-floating self IP, both will process DNS requests (but you need to have both as DNS server for your zone).