A Set with Forbidden Sums

Let $E = \{1, 2, 3, \dots, 200\}$ and let $G = \{a_1, a_2, \dots, a_{100}\}$ be a $100$-element subset of $E$ with the following two properties:

1. for all $1 \leqslant i \leqslant j \leqslant 100$, we have $a_i + a_j \neq 201$; 2. $\displaystyle\sum_{i=1}^{100} a_i = 10080$.

Prove that the number of odd numbers in $G$ is a multiple of $4$, and that the sum of the squares of all elements of $G$ is a fixed constant.