Question $3$ is the only one that is connected to long-established theory.
$1105=5 \times 13\times 17$, and that each of $5$, $13$, and $17$ is easily seen to be a sum of two squares.
By an identity that is a special case of Brahmagupta's Identity, the product of two sums of two squares is itself a sum of two squares. The identity is
You can use the above identity to generate the representations of $1105$ as a sum of squares. Do it!
The identity is interesting in many ways, and is less "magic" than it looks.
Let $p+qi$ be a complex number. Then $|p+qi|^2$, the square of the norm of $a+bi$, is precisely $p^2+q^2$.
The Brahmagupta Identity then can be interpreted as saying that
$$ |(a+bi)(c+di)|^2 =|a+bi|^2|c+di|^2$$
(the square of the norm of a product is the product of the squares of the norms, or, taking square roots, the norm of a product is the product of the norms.)
Expressing $1105$ as a sum of two squares can be thought of as finding all the ways to express
in the form $(a-bi)(a+bi)$, where $a$ and $b$ are integers.
There is a quite thoroughly worked out theory of sums of two squares, that you can find in most books on Elementary Number Theory. For example, a prime of the form $4k+1$ can be expressed in essentially one way as a sum of two squares. This is a result of Fermat. The proof, though "elementary", is not all that easy.