The scene I previously analyzed was from Episode 5 of Cowboy Bebop entitled Ballad of Fallen Angels.
The main named characters were Spike and Vicious. The scene suggested they had a history, and a falling out. It also suggested that the moderately-clad woman and gruff bonsai-tending man were also primary characters in the show.

Firstly, In my search for more information on Cowboy Bebop I’m going to consult my go-to reference: Wikipedia. I’ll be back in a few minutes.

Twenty minutes later:
Phew! Little did I know I was missing out! As soon as I get home I’m going to borrow this series and watch it in its legendary entirety. Whenever I’d heard the name “Cowboy Bebop” out of context, it sounded cheezy. I’m not fond of westerns and a bebop sounded childish. But I was wrong about My Little Pony: Friendship is Magic, so why not an unusually-titled anime legend? On the subject of legendary, Yoko Kanno did the music, wow.

The sequence I analyzed was referred to as “one of the most influential scenes in anime history” which makes me feel like I chose in good taste. I was also correct in my assumption that the large gruff man and mildly-clad woman were main characters: Jet Black and Faye Valentine, respectively. After reading about the petite hacker and smart Corgi, the main characters who did not make an appearance in the clip, I’m quite excited to watch this show.