Q1. In my
opinion, Dmitry's record will stand for a very long time !
Q2. I think long runs are not especially obtained by
numbers of the form 2^n + 1.
Let us denote the maximal number of consecutive integers
with the same Collatz length as n by L(n).
I found 1158 values of n < 2000 for which L(2^n + 1)
For 3 < n < 22, the maximum of L(m) when 2^n < m
<2^(n+1) is never obtained when m = 2^n + 1. But this can
be due to the fact that this deals with small numbers.
Q3. I tried numbers of the form 3*2^n + 1 for n < 468.
I found L( 3*2^n + 1) = L(2^n + 1) with only three
exceptions : n = 17, 67 and 68.