Why is a ~cover~ mentioned in ~please bring a covered dish~?

It's just a phrase. Which doesn't exist in British English.

When I used to go to church (England here too), our church was good at this and did it a lot. It's that church's knee-jerk response to catering at any social event. "Please bring food to share" is always what was said, or it was called a faith lunch if at lunchtime, and everybody knows what it means. Bring some PROPER food, not just something you bought. So it works. It just seems so obvious that food should be transported covered that we never said it.

Another good phrase is "faith tea", and you'd know that means salad, sandwiches and cake. Another thing we did occasionally, being a London suburban church with all colours and nationalities, is have an International Evening with "bring food to share". The obvious implication is "bring something from your homeland".

"Covered dish" and "potluck" are just American phrases for the same thing, and bear in mind American English isn't British English. Pot luck makes sense - you take pot luck as to what you'll actually get! Really it's just a matter of we don't speak the same kind of English as Americans.

It usually refers to a hot dish, like casserole, soup, pasta.

You could probably get away with bringing salad, dessert, or drink.

The health department requires dishes to be covered.

People would bring beverages or ice cream and such stuff what it means to bring a proper , cooked dish.

a casserole dish of some kind

"Covered Dish Event", "Potluck", same thing.

Why not "please bring some food to share"? Well, I bought a bag of chips. Does that count?

Potluck/Covered Dish implies an actual prepared dish of some sort, preferably homemade. Salad, entree, dessert, etc. - not just a bag of snacks.

For food hygiene. A foil or film cover is sufficient, a lid is also acceptable. It keeps flies and other contaminants from the food and reduces the risk of food poisoning from cross contamination at the event setup or during transport.