Can we 'see' the same distance in every direction?

Just like the 2 dimensional surface of the Earth is curved in the 3rd dimension, the 3 dimensional space of the universe is believed to be curved in a higher dimension that we cannot see. You cannot say that anyplace on Earth is the center of the Earth, or the edge either. The center in deep inside the earth, outside the 2 dimensional surface, and the edge is the surface you are standing on. If you go south long enough, you will come to the south pole. If you keep going in the same direction, you will be going north, because every direction is north if you are at the south pole, and eventually you would pass the north pole and finally come back to where you started. You might think you went in a straight line, because you never turned left or right. But you path curved DOWN the entire time you were moving. Your path curved in the 3rd dimension. In an analogous way, if you went straight out from Earth, not turning left or right or up or down, you would really be curving in another dimension, in a direction we have no name for and cannot see, but only describe with math, and possibly you could come back to Earth. So Earth is not in the center of the universe. The center would be in another dimension and we are always at the edge of the universe, no matter where we are. Strange, huh?

Our ability to detect distant objects is limited by the size or brightness of the object as well as dust or other obstructions in space. The current distance we can detect is more of a limit of our capabilities than an indication that we are in the exact center.

Essentially, yes...barring clouds of dust and such that get in the way of seeing behind them. Your quandry has also bothered me for a long time...though Einsteinian physics has a rather twisted explanation for this observation. (As I'm sure others will attempt to explain.)

The latter of your two suspicions is correct, get yourself a formula

toj compute infinity and, Presto ! you'll know the size of the universe.

Actually, no. There is dust and gas blocking our view to one side of our galaxy.