5.You can check if a number is divisible by a number d with a n-digit sum if there some m such that m*d=10^n-1. So for 11 it would be 9*11=99.

2.And this even works almost the same if you can find a multiple that is close to a power of 10. For example 11 is 10+1 so you can check divisibility by 11 by doing a alternating 1-digit sum:
3-0+8=11 so 308 is divisible by 11
4-5+3-8+2-5=-9 not divisible by 11
1-8+7-8+6-9=-11 divisible by 11

For 17 you can check divisibility using 2-digits because 17*6=100+2.
So if you wanted to check if 165616 is divisible by 17 you would divide 165616 into 2-digit pairs and starting from the left subtract 2 times the digit pair from the next pair. Also subtract multiples of 17 if possible.
16 56 16 -> 24 16 -> 7 16 -> 2
11 86 60 -> 64 60 ->13 60 -> 34 -> 0