Is this a live game? (in General)

When I click the "fight" button, am I playing against another person playing or against the computer?

You are fighting another character, the way it was at the time of you clicking "fight". There is no interfacing with a real person though, the game handles the fighting for you. you are more of a spectator, viewing the fight results.

Now, 5 seconds, or 2 minutues, or 5 weeks after you fight that guy, he coudl upgrade his skills, or train experience, or upgrade his weapons, and he will be harder to beat next time you click on the "fight" button.

Every character in the game is human created. As far as it being 'live', not so much. The combats are automated and computer 'refereed'. Combat follows a 'script' as to what can be done in each round. Any of your minions that can act in a round will act in that round.

You (and your opponent) control what your character's minions do each round by choosing spells and equipment ahead of time. That's really the only way you can influence what actually happens in combat.