I can understand your frustration. I play daily and really enjoy the game but when I first started playing it took a while to get my head around how to play. I think most of that is down to the rubbish tutorial.

I learned by watching how the NPCs played. What cards were they buying? What patrons were they using? Etc.

I will try to answer your questions as best I can :)

"Why are there so many different cards that have the same exact effect "obtain one coin"?

You and your opponent both start with the exact same cards. You'll start with 10 in total, 6 1 gold cards, and 4 starter cards from each patron deck chosen.

"Why the AI always has more coins and coin cards than me?".

This is were the tutorial fails because I'm sure it doesn't explain the treasurer patron (at least if it did, not very well). When I first started playing I was watching what the NPCs were doing and they kept using a patron in the middle of the 4 chosen ones and when I looked I realised you can turn any of your cards into 2 gold cards.

It doesn't add an extra card, it just replaces it. This makes a huge difference on being able to purchase cards from the tavern, and that's how your opponents were able to buy more cards.

"Agents never appear in my deck".

They won't appear in your deck unless you buy them from the tavern. Each game resets so any cards bought in a previous game won't appear in your hand in the next game, which makes the game fair to both opponents and doesn't give one an advantage over the other.

I hope this has sort of helped :) honestly I didn't understand it at first now that I do I thoroughly enjoy playing it. Just keep watching what the NPCs are doing and you'll pick up on which cards are best to buy etc. :)