Rather than make numerous questions on the forums, figure to keep my questions here to you! :).
Does the Allied AI surprise a human German player and conduct amphibious landings other than the scheduled Husky and Dday?
I'm going to have to plead ignorance on this one. I've only ever played the Allies.
Do construction units attached to cities speed up the building and repairing process occurring in said city? (i.e Palmero is captured by allies, damage to port:99. Does a construction unit repair that and help with building/expanding an airfield?)
Construction units DO speed up repairs and building airbases. They prioritize repairs over the airbase. You can assign multiple construction units to a city, which further speeds things up, though there is some maximum amount of speed-up that I don't know how to calculate.
A best practice for construction units is at the end of every ground phase before you end your turn, run the Commander's report ("c"), filter to only Construction units and only Support rather than on-map, then sort by HHQ. You can then click on each, see progress, start another round of airbase building, unattach and then reattach to another city, and also see any that the AI has put onto the map (it will show a number under the "DtHq" column), and possibly reassign those or return them to your HQ.
During the check phases of battles, are units stats and damage really taken into consideration? I meticulously planned a very strong counter attack in Italy using my Tigers and Panthers vs an allied weaker position but still filled with lots of Sherman IIIs and M4A1s. Lost, and with that many tigers and panthers... No air superiority and a few arty units passed their checks and came into battle on the allied side.
It is supposed to. Did you look at the after-action report by clicking F11? One of the tabs in Show Detail shows Ground Combat.