Yes, it is a very formal, and, posh phrase :P I personally can only think of the following situations: (1)a waiter/waitress saying 请慢用 to the customers when the food is served (I think this is the most common situation in whicn you would hear this.) (2)to important guests when you treat them at a fancy/posh restaurant, when the food is served and you would like to suggest starting the dinner, such as to your business clients etc. (3)I agree with Wang Haoyu. If you have to leave while others are still having dinner, especially when you are not familiar with them/meet them for the first time, you could use "不好意思，我得先走了，你们请慢用" to tell them that they don't need to hurry, just enjoy the meal and you feel sorry to leave earlier. I would not say this to those who I'm very familiar with. For example, I would never say this to my parents no matter where we have dinner. I would never say this to my friends either. Or I could, if I wanna be humorous, I guess. This phrase suggests great respect to the guests. However, such respect can imply that you are not familiar with each other yet.That's why there is no need to say this between friends and family members.

You are right. 请慢用 and 慢用 are used for feast and banquet with friends relatives. or eating out at a restaurant. at home, at school, any places can apply. formal and causal cases are ok too. usually, waitrons use it politely when among family and friends they are good etiquette.

It can be used both in restaurants or guests (uncommon guests or elders) at home to show courtesy. However, using it on close friends/relationships might sound a bit weird, in which case, you might want to use 快吃吧/开动吧 (Let's eat).

You can use it anywhere. Just shows courtesy. Literally, "use it at your leisure"

我觉得“慢用”和“请慢用”的意思和enjoy your meal还是有区别的。比如，我们和朋友在饭馆吃饭，吃到一半，我有事情必须马上离开，这时我就可以说，我先走了，你们慢用。其实在饭馆里服务员如果对客人说，菜上完了，你们请慢用，其实更多的是在说“你们慢慢的吃，不用着急，我们不会催你们快吃快离开”。如果要强调“享用”，你可以听到人们会说“请慢慢享用”，也就是说，会把“享用”直接表达出来。所以“慢用”更多的是在表达“不用着急慢慢吃”的意思。