Batteries: It's worthwhile to get a "good" battery. The junk they sell at Wallmart, etc. is very sensitive to vibration and impacts. Harris sells good stuff if you have an outlet around.
R/R's: They do get hot, and that's normal. Its actually better to have a substantial load on your system to take some of the load off your R/R. So if you've done something to lighten the load, like exchange regular headlights with LEDs, then that can cause overheating. Updating to a better type R/R is worthwhile, but leave that install to someone with experience.
Connectors: Disconnect them, inspect for burn marks/overheating, clean them with a contact cleaner, then coat them with Dielectric Grease before reassembly.
Volt/Ohm/Amp meters: Careful! When set to Ohms, there is a direct short from red to black, so to clarify, do not check "VOM continuity across the battery leads..." in the "OHM" setting....use DC VOLTS setting.
When checking the phases on your stator the "Should be less than 1.0 ohms" can be influenced by your meter and the meter test leads. So, it may show a little more than 1.0 ohms.
Good luck.