Kosher in a secular hostel

Peace and blessings,
Good month!
 
I was hired as a guide at a hostel where students with cognitive disabilities stay. The students have different levels of disability, and different levels of religiosity – some are religious and some are secular. The management is secular.
Among other things, I am occasionally responsible for cooking meals, together with the residents.
Officially, kashrut is maintained on site, with the intention being that the policy is to purchase products with some kind of kosher seal.
In terms of separating meat and milk, I don't know if anyone there can be trusted. The guides are secular, and the residents, of course, can't be trusted. It's possible that they cooked milk in a new-born meat pot, or vice versa, and so on. The cooking utensils are not marked at all, but they usually don't cook dairy.
I guess I can't eat the meals cooked there (?), but I would like to ask about the actual preparation –
Can I taste the food during preparation to check the seasoning?
Can I cook in the dishes or in the oven without worrying that I am using non-kosher utensils?
What about donations and tithes, and about insects in food?
What about the issue of immersing the vessels?
And what about the trainees failing in matters of kashrut?