Skipping parts of davening - Igrot Moshe Orach Chayim 4:91

Can a teacher skip parts of davening if he is running late in the morning? In this teshuva, Rav Moshe Feinstein says that in such a case, the teacher can most definitely skip פסוקי דזמרא. He offers two rationales for this. First, if he were to come late to class he would be taking away learning time from his students; second, coming late to class would constitute stealing from his employer, who is paying him to be at work at a certain time. Rav Moshe notes that if the issue is that the teacher has his own work to do he should not skip any part of davening.

Rav Moshe does note the option of saying הביננו, which is a severely shortened version of שמונה עשרה, but concludes that since it is so rarely, if ever, said these days it is unlikely that the teacher will know it and thus the time saving as a result is likely to be minimal.

Tefillin on a dead arm - Avnei Nezer OC 1

The Avnei Nezer was asked about a person whose left arm had lost feeling and whether that person should continue to put tefillin on that arm.  By comparison, a bird whose wing dries out is considered to be lacking that limb - would we say the same by a person, that the useless arm is considered to not even exist, or could the individual still put tefillin on given that the arm does physically exist?

The Avnei Nezer differentiates between birds and people on this point.  He notes that by a bird, the wings are the main part of the animal (based on עוף יעופף), and that such a blemish would be considered a particularly catastrophic blemish, thus invalidating the animal.  Such an injury by a human would not rise to this level.

Another consideration is whether putting tefillin on the right arm would be considered better than putting them on the damaged left arm.  The Avnei Nezer feels that since the left arm still exists, there is no reason to place the tefillin on the right (wrong) arm.