Should I assume that the sums $a_i+a_j$ are distinct? If so, how about $a_1 = 10,a_2 = 10^{10},a_3=10^{10^{10}},\cdots$?

if the sums are distinct then the problem is solved. in your example the answer is NO because each sum is congruent with $ 2,10 $ modulo $ 100 $.

my first idea: let take the different residues from the initial sequence and note them with $ r_{1},r_{2},....,r_{t} $. then are at most $ t(t-1)/2+t=t(t+1)/2 $ different residues in the set of the sums. so $ t(t+1) \ge 200 $ so $ t=14 $ so all the initial numbers are different modulo $ 100 $. let's see how to go on!

No.Just consider the combinations with odd residues.($01,03,05,\cdots,99$) Let $r_1,r_2,\cdots,r_k$ is odd residues of $\mod 100$ of $a_1,\cdots,a_{14}.$ Similar, $s_1,s_2,\cdots,s_k$ is even residues of $\mod 100$ of $a_1,\cdots,a_{14}.$ Then the number of odd residues of $a_k+a_l$,$\le kl \le 7^2=49$.