By Tim (the typer)

and Sarah (the thinker)

Is the infinite sequence of all infinite sequences countable?

It would be possible to imagine pairing each with a natural number, but you would never be able to physically do it and finish. If you have every sequence, you have to have an order. There are infinite orders, so you can only call this 'an' infinite sequence of all infinite sequences, not 'the' infinite sequence of all infinite sequences as there would be an infinite number of infinite sequences of all infinite sequences.

You cannot make an infinite sequence of all sequences because there is always a sequence that you haven't got - you can find this by taking the first term of the first sequence, the second term of the second sequence, the third term of the third sequence, the fourth term of the fourth sequence and the nth term of the nth sequence and changing it in some way e.g. by adding 1, or dividing by 670009. You will not however then have all the sequences because there will be lots of other sequence like this.

Is the sequence of all sequences of digits countable?

The  answer is the same as above

Is the sequence of numbers between 0 and 1 corresponding to sequences of digits countable?

It would be possible to imagine pairing each with a natural number, but you would never be able to physically do it and finish.

If the sequence of numbers between 0 and 1 are not countable does that mean there are more numbers between 0 and 1 than rational numbers between 0 and 1?

There are more numbers between 0 and 1 than rational numbers between 0 and 1. There are some irrational numbers. For example π/4. This is irrational because π is irrational. If you multiply a repeating decimal number (like 1/3) by 4, it will still be a repeating number so you can't multiply an irrational number by 4 and get a rational number. So this works the other way; if you divided an irrational number by 4 it can't be rational.

Extra credit. Are there numbers between 0 and 1 that cannot be described with a finite number of words?

It depends how accurately you want to describe them. You could just say 'it is a zero point then the digits zero to nine repeated loads of times never repeating.' Some of the numbers have shorter descriptions that give exactly what you need to know to find them. Even if it doesn't give you the number itself. For example, π cannot be described simply, but there are ways to describe how to find it. One is to divide the area of a circle by the diameter.