Well, in my case, the order of operation in uni wasn't brought up even once. But it was also a prestigious one with notoriously challenging math courses, so I may be a little out of touch in that regard. (Let me brag, ok?)

No, I do not think those conventions are needed. Because they aren't fundamental. You don't really know math until you understand how PEMDAS or w/e came to be and why it is the way that it is.

Not following those conventions doesn't automatically make your solution incorrect. That's the most important thing.

It shouldn't matter how you solve as long as it is a correct solution.

There may, indeed, be inconsistencies in how things are written out. Whether 2x is the same as 2×x, for example. It's common practice that it isn't, but it's also often not important.

If you write out the solution, people will understand what you mean by simply following it.

Compare:

6÷(2+4) = 6÷2(1+2) = 6÷2÷3 = 1

And

6÷(6÷3×(1+2)) = 6÷2(1+2) = 6÷2×3 = 9

They are written in the same manner, but those are 2 different equations to begin with, with their respective correct solution. For the same reason why 2x and 2×x may be the same or not. (Replace 1+2 with x, you'll get 6/2x vs. 6x/2)

It's not a matter of order of operations, but a matter of context. Whether juxtaposition took place or not. In real research, 2x always has a context.

Besides, the equations aren't usually written out that way, aren't they? You would do this (except for the dot in multiplication, unless it's needed)