- Does anyone have a source for this (or for the opposite: that he put on his shel yad standing)?
- If he did put it on sitting, does anyone know why he did so?
I don't know whether he personally did, but he does mention this custom in his siddur, and gives the Zohar as the source. (It states that the seated part of davening corresponds to the shel yad, and the standing part - primarily Shemoneh Esrei, I guess - to the shel rosh.)
Dayan Raskin, in his notes to the siddur there, has an extensive discussion of this issue, and cites a source that R. Shalom Dovber of Lubavitch indeed followed this practice. (Present-day Chabad custom, though, is to put both tefillin on while standing.)