Learn English – Formal way of asking- “what’s your name”

When you seek a direct object, you typically ask for something. However, I would use neither of those words. The best words are found in Maulik's post, though for some reason he did not state it. When sending out invitations, use request. It's very common, polite, and not seen as begging, while asking/seeking for something does.

Saying "I ask for the pleasure of your company" almost sounds as if you're begging them to come, and not in a good way.

The most authentic-sounding is the second.

Which room are we meeting in?

Option 1 is not valid. You are meeting (I imagine) in the room, and not "at" (i.e. outside it).

Option 3 is valid but likely overly formal.

Option 4 is not grammatical. "Where" and "in" do not play nicely together. When combined like this, they become "in which". So, you could potentially say (though I maintain it's formal/archaic): "Where/Which is the room in which we are meeting?".