Lots of questions and i skimmed through them but here is my input:

1. You can download from hptuners repository a stock 2008 Hummer H2 file. Im not sure whwt ecu the H2 used but if you use a third party tuner (pcmofnc) they can easily adapt the H2 tune to your H3 E67 format (I had them do that for my ls376/480 swap into my alpha). It will put your tune at 95% right and will sort out any dod/vvt variations assuming the harness is right

2. id recommend you use the H2 injectors if they have been swapped. Otherwise you will need to supply the correct alpha injector data to the tuner (the alpha tune should have it right there)

3. The e67 ecu can easily run efans, takes 10 mins for the tuner to set all the tables. I recommend a camaro ss fan if you want to get the best fan there is (cools my 500 hp easily in 125* weather) and its pwm meaning speed variation from 15% to 100%. As for fan logic, if you are using a dual fan setup, check the 2008+ H2 tune and copy it. Just wire in the dual fan using the GM 3 relay diagram. If you go with pwm, check the corvette/camaro or newer truck logic and adapt it to the fan control tables in hptuners (its straight forward)

4. Changing the gear calibration to 4.56 from 4.10 will sort out the trans issues. Keep in mind you need to address the tires too.