  We have measured the doping and temperature dependence of lattice vibrations
and magnetic excitations in the prototypical doped spin-orbit Mott insulator
Sr$_{2-x}$La$_x$IrO$_4$ (x=0, 0.015, and 0.10). Our findings show that the
pseudospin-lattice coupling -- responsible for the renormalization of several
low energy phonon modes -- is preserved even when long-range magnetic order is
suppressed by doping. In our most highly doped sample, the single magnon
(Gamma-point) excitation disappears while the two-magnon mode softens and
becomes heavily damped. Doping induced electron-phonon coupling is also
observed in a higher energy phonon mode. We observe two different
electron-phonon interaction channels, which provide evidence of the coexistence
of fluctuating magnetic moments and mobile carriers in doped iridates.
