Shulchan Aruch and Rama, Yore Dea 115:2, indicate that cheese made by a non-Jew can't be eaten unless a Jew saw it being made. Kraft, a national brand, now has some Ⓤ-certified reduced-fat mozzarella cheese. Is a Jew present constantly, or is there some reason he doesn't have to be?
You can ask the OU to confirm, but yes, they have a Jew present constantly when the cheese is being made.
They may have some days where the cheese is not Kosher and they have arrangements to have that sold as non-Kosher.
Yes, that is expensive. Next to meat, it is the most expensive form of Kosher certification.
The supervision provided by most kashrus agencies normally fulfills both opinions, such that when the mashgiach is present for cheese production (thereby fulfilling the Remo’s requirement), he personally adds the rennet enzyme to the milk for each batch of cheese (thereby fulfilling the Shach’s requirement)
Common practice (adopted by many kashrus agencies, based on the rationale in Igros Moshe YD 2 s. 45 and approved by Rav Yosef Eliyohu Henkin) is to follow the opinion that that the gezerah of gevinas akum was only declared on hard cheese, as only hard cheese uses rennet to form into curd and was therefore subject to the prohibition.