The air intake is there to bring combustion air in from the outside, that's it. The theory is (and it is debatable, but still law) that there is not enough oxygen in the home to support both combustion and you breathing.
The black double wall stove pipe connects the stove to the chimney. The chimney comes through the roof and uses an adapter. A ring attaches to the bottom chimney section and goes through the adapter the chimney pipe screws onto this ring. Here is the link to the manufacturer I used. You can look through the installation instructions there.
I used the secure temp 2100 chimney that is rated to 2100 degrees F and the DL chimney pipe. I think that the 2100 is not available in the US so the ASHT chimney would be used there. The DL pipe was only one extendable section and the adapter to attach it to my stove. 2 sections of chimney, the cathedral roof adapter and flashing were all the same company. This is one job where there is no such thing as overkill on safety. I don't want to burn down my house, lol. I have a few pics if you need so don't hesitate to ask.