Legs Eleven

Choose any two numbers from the $7$ times table. Add them together. Repeat with some other examples. Notice anything interesting?
Now do the same with a different times table. What do you notice this time? Convince yourself it always happens.

Choose two digits and arrange them to make two double-digit numbers.
For example, if you choose $5$ and $2$, you can make $52$ and $25$.
Now add your two-digit numbers.
Repeat with some other examples.
Notice anything interesting? Convince yourself it always happens.

Look at this sequence of numbers: $11, 101, 1001, 10001, 100001, ...$
Divide numbers in this sequence by $11$, WITHOUT using a calculator.
Notice anything interesting? Convince yourself it always happens.

Take any four-digit number, move the first digit to the 'back of the queue' and move the rest along. For example $5238$ would become $2385$.
Now add your two numbers.
Is the answer always a multiple of $11$? Can you convince yourself?
What happens when you do this with three-digit numbers? Five-digit numbers? Six-digit numbers? 38-digit numbers ... ?
Prove your findings!