Answer: 312211
John Conway's "look and say" sequence has no digits other than 1, 2, and 3 in it, unless the seed number contains such a digit or a run of more than three of the same digit. A degenerate sequence appears with the seed of 22.

Answer: 91

Answer: 3
Of course, as time passes, the smallest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration.

Answer: No, it's not!
Adding up the divisors of 36: 1 + 2 + 3 + 4 + 6 + 9 + 12 + 18 = 55

Answer: 5
Portman collaborated (using her birth name, Natalie Hershlag) with Abigail A. Baird, who has a collaboration path (Gazzaniga, Michael S. -> Victor, Jonathan D.) leading to Joseph Gillis, who has an Erdős number of 1.

Answer: Yes, it is!
Adding up the divisors of 8128: 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

Answer: 3
Russell collaborated very little and no Erdös number can be found using research articles. But, as pointed out by Sachi Sri Kantha at "the Erdös number project" Russell and Albert Einstein did coauthor the Russell-Einstein "Peace" Manifesto of 1955.

Answer: 333

Answer: Infinite

Answer: 49