I am not that fluent in Modulus, but I will give a try,
1) Each week has 7 days. Since there are 77 days, summer break will start and end on the same day.
Summer break starts on Thursday (1)
and lasts to Thursday (77).
So the day school starts should be Friday, the next day.
2) He starts at 12, assuming you made a typo in your sequence. "11, 23, 34, ..., 1233" should be 12, 23, 34, ...., 1233.
So since he started at 12, the sum of all the terms divided by 11 will be the same as if 12 was divided by 11.
If his sequence started at 11, the sum of all the terms in the sequence will be a multiple of 11.
Since it started at 12, the remainder is
12 / 11 + 1
The remainder should be 1.