Direct sum of Binary numbers In Mixcolumns
You misunderstand $(02) \cdot 10000100$; it is not integer multiplication (resulting in a 9 bit integer); instead, it is multiplication in $GF(2^8)$ (which results in an element in $GF(2^8)$, which can be represented in 8 bits).

AES uses a polynomial representation of $GF(2^8)$, using the polynomial $x^8 + x^4 + x^3 + x + 1$; what this means is that multiplication of two elements can be performed by multiplying the two values (however, disabling all internal carries); and then reducing the product modulo $11b$ (again, without doing any carries).

When we multiply by 2, this can be simplified to "shift all the bits left by 1, and then if the original bit 7 was a one, xor in the value 0x1b, and take the lower 8 bits of the result".

Given that, multiplying by 3 (which is also used as in the Mixcolumns operation) is easy, as $3 \times x = (2 \times x) \oplus (1 \times x)$, and we know how to evaluate both $2 \times x$ and $1 \times x$