First, I don't think it's true that "flat implies infinite." (There is a 14-upvote answer on the linked stackexchange question that mentions this problem -- the "flat implies infinite" argument implicitly assumes that the universe has the topological property of "simple connectedness," and I see no physical reason other than convenience to assume this.)
This is only an answer to question one, "what does flat mean in 3 dimensions." I have to talk about math some, but I won't be too formal.
Imagine that you live on the surface of a sphere (this shouldn't be so hard to imagine, because you do.) If the sphere is very large compared to you (like it is) then you might think, just from observing your local area, that you are actually living in a plane (which is what the ancients thought).
The sphere is an example of a 2-manifold, which is a shape that "looks locally like a plane" i.e. a sufficiently small inhabitant can't tell it from a plane. Another example of a 2-manifold is the plane itself. Yet another example is a "torus," i.e. the surface of a donut. I hope you'll agree that the plane is intuitively "flat" and the "sphere" and "torus" are intuitively not.
It is very important to distinguish two different branches of mathematics that talk about 2-manifolds before we continue. These branches are "geometry" and "topology." In topology, we say that two two-manifolds are the same if one can be continuously deformed into the other. In geometry, which is more familiar, we say they're the same only if they are congruent.
Now I am going to describe a 2-manifold called the "flat torus." If you've ever played the ancient video game "asteroids," you've already seen it. This 2-manifold looks like your computer screen, but we agree that any object that leaves the top of the screen comes back up through the bottom; any object that leaves the right comes back in through the left. (Pac-Man had a similar logic and I think lots of old video games -- I hope you can picture what I am talking about!)
Now, topologically, the space that I just described is the same as the torus. That's because if we take a piece of paper and glue each pair of opposite edges together, we get a torus. But geometrically, it's different: it's flat! We can define this precisely (and measure it experimentally) by looking at the deviation from the sum of the angles in a triangle from 180 degrees.
Now of course the universe is not a 2-manifold. It's a 3-manifold. That means it looks locally like "3-space" everywhere (in the same way the earth looks locally like a plane everywhere). It doesn't follow that it looks globally like "3-space" (i.e. infinite in all directions) -- we could just be really near-sighted. How could this fail to be flat? Well, that's difficult to picture, but for example, the 3-sphere with its usual metric is not flat. The problem is that our normal way of visualizing 2-dimensional spaces that aren't flat (like the sphere or the non-flat torus) is bending them in 3-space, but we can't visualize what it would look like to bend a 3-dimensional space into 4-space.
However, it could be flat and still be finite. For example, you can imagine "3-dimensional asteroids", i.e., picture the room you're in, with the rule that if you pass through the ceiling, you come up through the floor, if you pass through the front wall, you come in through the back wall, if you pass through the right wall, you come in through the left wall. If the universe is a huge copy of that (geometrically and topologically) then it's flat and finite.
Of course maybe the universe is just an infinite flat 3-space. Perhaps this is Occam's razor, but humanity has been wrong before (e.g. by making this exact assumption for shape of earth). Or maybe it's not even flat, and our measurements are too imprecise.
Technicality: my understanding is that the universe is not a 3-manifold; rather, it is a 4-manifold ("spacetime"), but there's no product decomposition of the form spacetime = space cross time so it doesn't really make sense to ask what 3-manifold space itself is.
Recommendation: read the book "The Shape of Space." This is the second time I've recommended this book on reddit tonight!
tl;dr the universe is probably flat cause triangles look that way, but maybe our triangles aren't big enough. even if the universe is flat, we can't assume it's infinite without unjustified extra topological assumptions.