Thanking everyone for their helpful tips and comments. This was a roller coaster and which took a lot of research and work to sort through. But it was a fun project.
1. Alternator diode test
We ran the scope at faster rates (see image below: 10mV/div, 1ms with 10:1 probes).
--> There doesn't seem to be much AC ripple so guess that the alternator diodes are OK. Feel free to correct us here!
2. Parasitic drain test
We locked the car (closing bonnet alarm sensor with a plastic clamp). Then connected DMM between the negative battery terminal and negative cable (without breaking the electrical connection).
--> Parasitic drain around 20mA after about 30 seconds. We tried that test several times so assuming no abnormal parasitic drain
3. Cleaned a lot of grounds and the contacts at the battery, alternator, and starter.
Cleaning the grounds and contacts didn't really change voltage readings (we cleaned most of these terminals a few years ago).
--> THE main ground wire was corroded at the starter side (underneath the car). Both the ring connector and copper wires were green. Really dfficult access so home-gamer probably could not cut off green wire and professionally recrimp a new connector (or try serious cleaning in-situ).
--> Very low voltage readings from + battery terminal to BOTH chassis & negative - battery terminal.
--> Both cables at battery (+ and - cables) still were hot to touch after running a lot of accessories for 10 minutes.
--> Voltage drop testing from negative battery terminal - to chassis was NEAR ZERO. We were seeing above 0.5v at engine block, alternator body, etc. Engine idling with lots of accessories on. That was a good clue.
4. Purchased additional ground cable & split corrigated plastic loom tubing for protection. We used some white string to map out a path and to estimate cable length. Basically followed main ground from starter to battery terminal, using zip ties. Took a long time as the ground was big and inflexible; plus had to avoid coolant lines, sharp edges, etc.
Cable is copper, 1-AWG, 5-foot long, ring terminals from factory that we purchased from local wholesaler buddy; Standard Motor Products and he said cable was high-quality with good crimps. The car's factory ground was a bit thinner, so a thinner cable would have been sufficient and easier to work with.
--> Voltage readings at chassis and - battery terminal fixed
--> Battery cables no longer hot to touch
5. VOLTAGE TESTING data (after cleaning terminals & after adding addtional ground, respectively). We didn't have an extra person to hold the accelerator pedal at high RPMs so only have data at idle.
Car off
12.7v, 12.8v battery terminal + to -
Idling, no accessories
14.2v, 13.9v battery terminal + to -
Idling, lots of accessories (headlamps, radio, rear defroster, AC at high). Connected DMM to + battery terminal:
*13.1v, 13.8v negative battery terminal -
*13.0v, 13.8v chassis
13.9v, 13.8v valve cover bolt
13.8v, 13.8v engine block
13.8v, 13.8v alternator body
*68A , 81A Amp clamp at positive cable out of alternator (We didn't have the amp clamp before cleaning)
CONCLUSION - We are guessing that corroded main ground wire prevented proper battery charging & caused hot battery cables. We still need to do some final work (e.g. voltage drop testing, put car on scope when starting to see lowest voltage, etc.) We will follow up with any additional findings.