AOT: you decide who you want as Corps Commanders from those three and set up your Corps the way you want them.
You can only form divisions with an active leader. Each turn, each leader makes a check against his strategic rating (the 3 in a 3-1-1 general) to see if he is active or not. Inactivity gives movement and combat penalties, and prevents him from executing offensive orders. I thought they couldn't form Corps either, but if Dodge can, then....
In the Grand Campaigns, you get a new infusion of generals at the beginning of every year, but I don't know about the 1862 scenario. Each scenario has a script that determines when and where units come on the map. For a definitive answer, you can look in the game files for something like a Scenarios folder and dig through them to find it. They look weird, but the language is pretty readable by non-programmers (like me). I would look for you, but I no longer have AACW installed as I recently switched computers. Otherwise you just have to play through once so you know what is going to appear and when.
The unit name XXXX's Brigade is a flavor only title in AACW and CW2. Hooker isn't necessarily attached to the brigade (unless you see his portrait on it) and other people can command it. As to whether Hooker himself is going to show up, I can't say.
If your divisions are not full already (17 elements plus the leader) then you won't need to form new divisions, you can add them to the ones you have. I don't know how much building of fresh troops that scenario allows, but you may simply be constrained by the number of leaders you have (there is rarely excess leadership in AGEOD scenarios).