If I have the sum of 3 or more numbers, e.g., 1 + 2 + 3 + 50, will the sum end up being done in pairs, i.e., the + operator is binary and thus only supports 2 operands at a time?
I am new to algebra, I'm talking about basic arithmetic, the basic addition that we all know.
Yes, the addition operator is purely binary. However, because it is associative (meaning that $(a + b) + c = a + (b + c)$) and commutative (meaning that $a + b = b + a$), it doesn't matter what order you perform the additions in and so $a + b + c$ has n unambiguous value, hence we omit the parentheses in typical notation.
This becomes important when you introduce other operators that don't have the same properties - or we work in a different system where $+$ behaves differently - and we need to establish a clear order of operations.