what kind of pump and attachment to tie my improvised shower to the sink drain?
You could use a "wye tailpiece" from a plumbing shop. These are normally sized for kitchen sink drains (1 1/2" tube) so you may have to adapt it (it can be done with creative use of drain fittings).



Use a submersible fountain pump, they are small and self-priming (please ensure that it is plugged into a GFCI). Just run a hose from the pump discharge to the tailpiece (again, may need creative adapting).