We normally use copper cladded rods, but zinc plated are available, and the differences are due to geographical/market locations.
The connection to the rod is usually just below grade, to help with insuring all 8 feet are in the ground, and helps with physical protection of the connection.The connector we use is called an "acorn" because of it's shape, has one bolt that tightens on the rod, and squeezes the wire onto the rod.
After thinking about it, I had to check, and seems I am imposing our local practices onto you.
A supplemental ground rod is only required if the main grounding electrode is a metal water pipe. So looks like your area is fine with one rod. This is what happens with me blending national code with my local code. Having it pounded in my head for 30 years, no pun intended, that two electrodes be used, always, whether it is a water line or a rod, I end up giving the wrong advice to others in different areas.
Living in New England, the land is all hills, mountains, and ledge, and we have a serious grounding problem here. So all local and state codes require AT LEAST two grounding electrodes, and sometimes more,no matter what. The resistance of any ground needs to be 25 ohms or less, and we have a real difficult time getting down to 25 ohms, let alone less.
Soil conditions have a great deal to do with good grounding. It is a real challange here trying to drive an 8 foot rod when there is ledge 2 feet below grade. And the soil can be dry because it is so rocky or sandy, and the water aquifer is down 100+ feet running thru fractures in the rock.
This is why local codes should always be checked. But if this is the worst advice I offer, all it costs is $10.00, and exceeds the national code.
So looks like your area is fine with one rod. If you were to install two rods, then the wire needs to be continuous from the neutral bar all the way to the last electrode. There is no certain depth the wire needs to be from building to rod to rod, just as long as it is kept below grade to protect it from physical damage. Again check with local code on this.
Yes use the one rod for all other grounds such as phone and cable.
If you like you can review the National Code at:
NFPA 70: National Electrical Code®
Chose preview this document, agree to the terms, and look for Article 250 Grounding in the table of contents.