Would one be allowed to use fingerspelling to communicate during pesukei d'zimrah (or other parts of davening where one is not allowed to talk)? Would it make any difference if one was hearing-impaired and always communicates using sign language?
The prohibition during Shma is in O"C chapter 63 law 6. See also Kitzur SH"A (17 6) that during the second parsha it may be permitted if needed for a mitzva.
See also Kitzur (18 14) that hand motions are not permitted during Shmona Esrei.
It is also prohibited to communicate via hand motions during bentching, and in between a bracha and the object of the bracha (e.g. after saying the blessing on tefillin until you finish putting on the tefillin shel rosh).
(This isn't an answer but it might have implications for one.)
The consideration of non-verbal substitutes for verbal communication is discussed in the G'mara in Gitin, which says that in some circumstances one who lacks the ability to speak can satisfy speech-dependent requirements through writing or even manual (or cephalic) signing. This leniency is not granted to those whose speech faculties are intact.