In "Father's in the cafe", "father's" is understood as a contraction of "father" + "is". "Father in the cafe" does not have a verb, so it isn't a full sentence, but "is" is a verb and is included in the second sentence via the contraction "father's". That's why the first phrase is incorrect and the second sentence is correct.
In a cafe, you sit at a table and a waitress gives you a menu, then takes your order. Then you sit and wait till she brings you your food. After you finish eating, the waitress is supposed to bring the customers the bill.
In a cafeteria, there is a line of people, who take a tray and go along a line and pick food and put it on their tray. For the hot food, there are servers. The people tell them what they want and the servers put it on a plate and give it to the person, who puts it on his or her tray, then moves on in the line to the desserts and salads and takes what he or she wants and puts it on the tray. At the end is the cashier where people pay for the food on their tray. Then they go to a table and sit down.
Cafeterias as described above can be found in hospitals, YWCA, colleges, schools.
At places like Dairy Queen, A+W, KFC, there is a menu on the wall and people go to the counter and tell the girl what they want. The girl sometimes asks "Is this to stay or to go". A very important question because then they know whether to put it on a paper plate or in a box. The customer pays and then takes the food.
Places like Dairy Queen, etc. are known as "fast food" places.
I've worked in a cafeteria and a cafe and so I know the difference. In both places, I had to wash tables. In a cafe, customers could ask for a book of matches if they wanted to smoke, but not in the cafeteria where I worked. Waitresses get tips but I didn't in the cafeteria, which was in a big hospital.
Another difference is that in the cafeteria, meals are only served during certain hours. For example, supper between 4 and 6:30 PM, whereas you can eat whenever you want in a cafe.
Because it's not saying that he is a cafe, only that he is in one. If you were to say "My father is a doctor" then you would use the dash to indicate that he is a doctor. The dash is kind of an indicator of the verb to be but it is not used in the present tense so instead they use a dash. I hoped this helped.
в literally just means "in" - на can also actually translate as "in" in certain contexts, as well. In Russian you typically just don't use the verb "to be" when describing things, even though there are two verbs I can think of that do mean "to be" (есть and являться). Есть is used typically only in the past tense (I was) or future (I will be) and являться is typically higher level/journalist/official speech that's not necessarily used colloquially in every day conversation.
True. The frequency of usage is another issue though. "Отец" is hard to find anywhere outside bureaucratic environment like paperwork, or conversations with officials, or in the context of mockery that mimics those things for fun, e.g. exaggerated strictness. I am not sure the usage of "father" has these features.
"Отец" is hard to find anywhere outside bureaucratic environment like paperwork, or conversations with officials,
What are you talking about?
Ever since I had grown up, I would almost certainly use "отец" and not "папа" in reference to my dad if I were telling anything about him to a third person who was not a family friend. "Папа" would be too sentimental IMHO. And it's not just me: my wife does the same thing mentioning her father, even to me. Moreover, I will call my dad "отец" in his face when I do not feel like adding any sentimental overtones to our conversation, i.e. not that infrequently.