Yes, the article is important. "Who is sitting opposite father" implies your father, whereas "Who is sitting across the father" implies the father of someone else. This sentence would make a lot of sense in the context of gossiping or discussing other people in the nearby vicinity. The English sentence also makes sense if you are discussing a priest and you want to know who is sitting across from him, though I don't believe they use "Vater" to refer to priests in German.
We just don't really say "in front of" to talk about the way people sit around a table. If someone sits opposite, it's clear that they are facing each other. If you say X is sitting in front of Y, the implication could be like that they're in rows in a church or something and are both facing the same way. The exception would be that you sit "in front of" a panel at an interview or something and (hopefully) you're looking at each other then - I think the idea there though is the different balance of power - that YOU are in front of THEM for them to assess. Round a table, you are in front of them as much as they are in front of you hence opposite or across from is more appropriate.
Why do they say "the father". I've seen this numerous times and it sounds so bizarre. Is it common to say "The Father" and "The Mother" when referring to your parents? Like "Go ask the Father" as oppose to "Go ask your father". In this case 'Our' or 'My' seems more like the norm in english, depending on who you are talking to.
Talking to a sibling: "Who sits across from our father?" Talking to a friend: "Who sits across from my father?" Talking to your mother: "Who sits across from father?"
Is germans "Who sits across from the father" an all encompassing version of the above?
If someone is referring to "the father" or "the mother" they are generally speaking of the father of someone else, often a child, not their own father. You can replace any of these examples with "the mother": Who is the father? Where is the father? Who is that with the father?
Also: Who is the father of the bride? The father of my children is recently unemployed and cannot pay child support. The father of Casey, my daughter's best friend, is a really sweet man.
I can go on with creating valid sentences that use an article before father or mother and still have it make complete sense in English. Many of them could replace the article with a pronoun, but it is not incorrect to use an article.