Seat reservations are always required on the Narita Express. It is highly unlikely that the trains get booked out on that route on August 8.

Shinagawa to Shuzenji on 10 Aug?

The ride can be done entirely by unreserved trains (local trains and unreserved cars on shinkansen). Therefore, seat reservations are not required.

Shuzenji to Narita Airport on 11 Aug?

See above.

The best route is by Tokaido Shinkansen and limited express train via Maibara (or Kyoto). The last connection leaves Tokyo at 18:50.

August 13 is one of the more busy days with respect to train travel out of Tokyo. You should try to make seat reservations as soon as possible. The morning seems busier than the afternoon

As of now, the direct hikari trains between Tokyo and Maibara in the morning (between 8 and 12) are already booked out or have only few vacancies, but all other trains still have vacancies. It might be easier to get seat reservations for nozomi trains to Kyoto, but the journey via Kyoto is slightly more expensive than via Maibara.

The trains from Maibara to Fukui are also very busy. About half of the trains are already booked out, especially in the morning and early afternoon.