Just a figure of speech.
Does doG have a mind?
Define mind. If you mean a "soul," whatever that means, the sense of self, thought, consciousness, then, yes. If you mean a physical brain with gray matter, nerves, neurons, and electrons connecting it all, obviously no, given what I stated before.
Is doG everywhere and nowhere and somewhere? What is existence wrt doG?
What/where/how is heaven, limbo, and hell?
Who knows. Heaven contains such delights that no eye has ever seen, no ear has ever heard of, no human mind has ever fathomed. Again, these are in existence outside of this physical universe.
Please define timeless eternity?
Not sure what you're getting at?
Is doG allowed to go to limbo or hell? If not, why not?
What do you mean by "go?" Get this Judeo-Christian tradition of God as a big, old guy out of your head so we can have a meaningful conversation. Heaven and hell are creations just like this universe. God is not like unto his creation or bound by its laws and properties.
Specifically, how does doG interact with our physically real universe? And by what means does doG achieve said interactions?
What is the source of the properties and laws by which our universe operates? Gravity, strong forces, weak forces, inertia, energy, what causes the physical behavior to be what it is? (Vibrating super-strings? In that case, what causes them to vibrate?)
doG circa 33 AD (because we humans need to put a face to such an abstract concept as doG because it makes doG real (e. g. pharaohs));
Not all of us. Thankfully, I am of a religion that purposely takes very concrete steps to control this human propensity to associate God's qualities with those of his creation. It does earn us a lot of flak from other belief systems, though.
doG circa 2,000 AD (because doG can appear to us humans in any form, after all doG is doG);
It is not consistent with the majesty of God to appear as one of his creations. God does not "carnate".