To answer the questions in your first paragraph, we think of the plane partly for convenience reasons, but also we remember that when we talk about “topological” notions they should be independent of some of the geometric particulars.
What I mean is, while I might think of a flat plane, like a chalkboard, and you might think of a curved sheet, like a cloth or whatever, anything that I call a topological property should still be true if I were somehow able to bend or stretch my chalkboard. Maybe I can’t talk about distance, but if I draw a circle it has an inside and an outside no matter how I adjust the chalkboard (this is a surprisingly hard theorem to prove, however!)
On the other hand, you might be concerned that another, equally correct visualization of the plane exists, which you can’t tell looks like bending or stretching a sheet of paper. In this direction, we know the plane is metrizable, and has topological dimension 2, and various other facts. So I don’t think there’s any harm in thinking of it as a sheet in three-space, as long as you remember you’re allowed to play with it
Edit: Oh shoot, I meant to answer in particular your last question: No. the topology of the plane is completely separate from whatever visualization you find convenient. Which is why I would suggest, rather than emphasizing a particular viewpoint, think more about how you can change your visualization of the plane without changing the topology.