Mid-infrared Er:CaF$_2$ laser operating on the $^4$I$_{11/2 →}$$^4$I$_{13/2}$ transition is developed. Its power scaling capabilities and thermo-optics (fractional heat loading and thermal lensing) are compared under pumping into the $^4$I$_{11/2}$ and $^4$I$_{9/2}$ states. Using a 4.5 at.% Er:CaF$_2$ crystal, a recordhigh continuous-wave output power of 0.83 W is achieved at 2800 nm with a slope efficiency of 31.6% and a laser threshold of 24 mW and the fractional heat loading is measured under lasing and non-lasing conditions, yielding the values of 52.0% and 71.7%, respectively (for pumping at 967.6 nm, into the $^4$I$_{11/2}$ state). The thermal lens in Er:CaF$_2$ is negative (divergent) owing to the negative thermo-optic coefficient and large and negative contribution of the photo-elastic effect. The sensitivity factors of the thermal lens are M$_r$ =-4.84 and M$_θ$ =-5.15 [m$^{-1}$ /(kW/cm$^2$)] and the astigmatism degree is as low as 6%. When pumping into the higher lying $^4$I$_{9/2}$ manifold, the thermal lens is enhanced owing to the additional heat generation from the multiphonon nonradiative path from this state, and the laser slope efficiency is deteriorated.