In Shinkansen's train plan, they tends to include every train paths. However, not every train will run on quieter days, leaving some paths unoccupied, which then might be used for test runs or "Dr. Yellow" (Shinkansen version of NMT).
Nozomi use 16 car N700A or N700S series (They operate all services on the Tokaido Shinkansen), while Mizuho use 8 car N700-7000/8000 series. In terms of the train categories and price, both of them are rank as express services, with the only difference being the the trains they use and the number of non-reserved coaches on the train. (3 in Nozomi, 0 in Mizuho)
In the case of Mizuho 620 and Nozomi 120, you might notice that Nozomi 338 is also using the same path between Osaka (10:39) and Tokyo (13:06). Nozomi 120 runs on 2/1, 11/1, and 25/1 in January, while Nozomi 338 runs on all other days. So this is a case of Nozomi 338 starting on Hakata as Nozomi 120 on certain days, probably as a relief service for the daily Nozomi 12 (Hakata 08:15-Tokyo 13:15). Then on 4/1 and 5/1, Mizuho 620 runs on the path between Hakata and Osaka, using 8 car N700, probably as part of a new year extra diagram.
Similar case for Nozomi 118 and Nozomi 336, but they operates less frequently, 23/11 only for Nozomi 118 and 2/1-5/1 for Nozomi 336. Looking at the timetable, both of them seems to be a peak relief service for Nozomi 10 (Hakata 08:00-Tokyo 12:57).
The timetable of Tokaido Shinkansen is based on a standard pattern, but out of the twelve Nozomi departing Tokyo every hour, only the xx:00, xx:12, xx:30, xx:48 are booked to run everyday, with the rest being extras. To make things more complex, some of the extra will have different destinations depending on the day and daily trains might be extended on busy days. For example Nozomi 221 (Tokyo 10:00-Osaka 12:30) being extended to Hakata and Hiroshima as Nozomi 129.
And since the train numbers are actually connected to the Headcode of each train, if the train are extended through Osaka they will need another Headcode as this involves the other company. (Train crew allocation, unit mileage, etc...)
Overall it is a very complex operation and their unit diagram seems to change every day, but it looks like both JRC and JRW are good at providing enough capacity for the demand.