Squirrel - dead or alive? 

Molly's memory of “that American that had the squirrel talking stamps with father” looks straightforward enough at a first glance, and yet gives rise to a number of questions that cannot be decided with absolute certainty.

  Excluding the possibility that Major Tweedy was talking stamps to a squirrel, we still cannot be sure if its American was male or female, if the squirrel was alive or a piece of furry clothing worn by the American, or (if it was a live squirrel) the American had it with him or her when talking stamps.

Squirrel fur (less likely). Squirrel as a short form of "squirrel fur" is certainly documented by the OED, and we may be intrigued by the fact that Joyce announced to Nora in a letter of 1 November 1909:

I hope to send you a splendid set of furs which I am selecting specially. They are grey squirrel. There would be a grey squirrel cap with violets at the side and a long broad flat stole of grey squirrel and a beige granny muff of the same on a steel chain, both lined with violet satin. Would you like that, dear? I hope I may be able to get it for you. Letters II, p. 258

  In addition, Molly is interested in clothes, so might treasure such detail.

  But unfortunately that reading would hardly make sense if the stamp-collector in question was male, as we would assume in company with the first translations of Ulysses which Joyce supervised: Morel has "cet Américain" and Goyert "des Amerikaners" (men wearing squirrel fur were the exception rather than the rule). Both translators take it for granted that the squirrel was alive and accompanying its owner on his visit to the Tweedys. Morel: "qui avait un écureul" and Goyert: "der das Eichhörnchen hatte". In both cases this would not work for an item of clothing.