möchte vs. würde gern when making appointment officially Hi. I saw the answer that möchte is bit more determined expression, and I understood it. If I write a letter for making appointment with city office, which expression should I use? Ich möchte einen Termin vereinbaren oder Ich würde gern einen Termin vereinbaren

I could use some company While I watch Matrix:Reloaded, I heard this phrase. When Neo came out in the middle of night and councillor Hamman came to him and asked: Care for some company? And Neo answered: I could probably use some company. And Hamman said: so could I. I think the phrase means 'do you need company?' and 'yes i do', but is it often used phrase in daily life?