As far as I can tell, the Checkmate server is a private, ranked server that you have to sign up for.
It tells you the website to go to to sign up, but I haven't bothered, I just use the YGOPRO servers.
They only have ones for EU and USA, but I've had no problems playing on the EU server from here in the US.
Add people how? Like a friends list or something?
I don't think they have one of those, although I could be wrong.
Adding people to a duel, you can only wait for them to enter your game.
I also find it weird/annoying that I can't activate Trap Cards during my opponent's phase without something to react to.
Not every Trap Card reacts to something, but it seems impossible to start a chain with one.
Plenty of Contiunous Traps, for example, can just as easily go off as chain link 1, but you can't do it.
A lot of the times they won't let me make a Trap chain link 1 until the End Phase starts even though there's no timing issues.
And, yes, that is a flaw that if you have simultaneous effects as the turn player, it usually forces the order even though you should have the choice.
That's what happens when you use an amateur made YGO game.
DN's problem was a lack of automation, allowing people to cheat or get rulings wrong,
and YGOPro's problem is that the automation isn't always perfect.
I have accidently interrupted the flow and it asked if I wanted to check if there was an effect I could use,
but it always seems to happen when I don't have something I actually want to activate.
The few times it's happened, I was either typing to talk, or I think I clicked at the right moment by accident.
One time it asked if I wanted to use an effect, and I thought it was an effect I forgot about so I clicked Yes.
However, there was nothing in my hand, grave, or on my field I could use, but since I clicked yes, the game wouldn't proceed
until I activated something. Luckily I was dueling the AI at the time, so I just quit the game since it was in limbo.
It's far from perfect, but what do you want for free?