A quick note in case anybody else runs into the same problem with GAPPS on the Nexus 5 (hammerhead).
I installed Dirty Unicorns 11 (Nougat 7.1.1) yesterday, and it worked fine.
Then I tried to install the recommended Dynamic GAPPS. It did not. The Mini version resulted in endless “Google Play Services keeps stopping”. Full Dynamic GAPPS resulted in the keyboard app refusing to run, making setup impossible.
So I moved on to OpenGapps, note that the hammerhead/Nexus 5 has a 32-bit arm processor, so do not use the 64-bit builds. The mini and full-sized builds would not install because of limited space on the system partition, and I’m too lazy to reformat everything. The “pico” edition did install, giving me the play store (which honestly is all I really wanted anyway, as I can bootstrap the apps I purchased there).
No success: Mini_Dynamic_Gapps-7.1.x-200170107.zip and Full_Dynamic_GApps-7.x.x-20170105.zip.
Note that I did a more-than-full wipe between all installs (data/cache/dalvik/system) before every install.
Hope this helps somebody searching for answers. The only bad thing about the Dirty Unicorns distribution is (not) finding information on Google Plus. :)