Spectrum finally sent someone out who was thorough. He ended up pretty much rewiring the entire house as well as replacing the line coming in from the utility pole and also changing the modem/router. The timeouts are completely gone. The DNS_PROBE_FINISHED_NO_INTERNET is completely gone. When I type in an URL and hit enter, it immediately starts loading without a spinning circle for minutes at a time. I watched streaming tv for 90 minutes and it did not buffer a single time.
Thank you, Thorough Cable Guy. You fixed my internet.