They indeed do become rarer. Plotting all the primes in a single row makes this apparent, like so: https://susam.net/primegrid.html#1-1-1000000

In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.

I have a small section about this at https://susam.net/journey-to-prime-number-theorem.html#prime... if you want to read more about this.

See also: https://en.wikipedia.org/wiki/Prime_number_theorem

Yes.

> In fact, according to the celebrated prime number theorem, the number of primes less than or equal to n is asymptotic to n/log n, which means the density of primes near n is asymptotic to 1/log n.

When written down as a string of digits, log n is another way to say 'proportional to the number of digits'.

The number of digits grows fairly slowly, thus also the 'probability' of a number being prime drops very slowly.