To see why the second fact is true, say $a$ has a remainder of $r$ when divided by $b$ and $c$ has a remainder of $s$. Then $a=bq+r$ and $c=bt+s$ for some integers $q$ and $t$, and so $$a+c=bq+bt+r+s,$$ which has the same remainder as $r+s$ since we can subtract multiples of $b$ without changing the remainder.

The third rule is similar: writing $a$ and $c$ as above, we now have $$a\cdot c=(bq+r)\cdot(bt+s)=b^2qt+b(qs+rt)+rs,$$ and again taking away the multiples of $b$ we are left with $rs$.

There is a Friday the 13th in October of year 1000000th?

Um. this would not exactly deal with that, but it seems likely, given that it is not an uncommon occurrence.

actually nevermind, in a specific month would be much less likely.