Obviously can't give you exact answers here, but for an engine at 100C (note: this is for the temperature of the outer wall of the engine, not the hot stuff on the inside) you get ~11W/m^2 of power emitted, which is pretty tiny compared to the massive heat flux you get from the air flow through the engine compartment. Say the engine is putting out 10kW of power at an efficiency of 10%. That means it's putting out 90kW as heat. Taking the surface area of the outer engine wall to be on the order 1m^2 (probably on a fraction of that, but close enough). I was lazy and picked easier numbers to work with (engine is probably putting out a lot more power on average, and the surface area is probably less), but that just goes to show how small 11W/m^2 is compared to the hundreds of kW/m^2 you'd expect from convection cooling.
Incidentally, the 11W/m^2 for the 100C blackbody you get out happens to be around a wavelength of 7.8 microns, which is inside an atmospheric absorption band (infrared near those wavelengths gets partially absorbed by water in the atmosphere). So not even all of that is going to make to the compartment wall to begin with, much less after a reflection. Granted, some of it can form a standing wave by incoherent addition, but the initial intensity off the engine is around ~0.01% of the convective heat flux. I just don't see it being more than 0.1%, in a worst-case scenario. If that kind of tolerance matters, then maybe it's worth taking steps to alleviate it.
As far as painting the walls of the engine compartment black, sure why not, if it doesn't cause other problems (paint peeling, rust, etc.). It would reduce the amount of infrared floating around in the compartment, but that level of radiation probably wouldn't matter in the first place. I doubt it's going to make the engine compartment act like an oven, because the engine compartment has a giant heat sink attached to it: the rest of the car, plus all the air surrounding the car. NASA has to worry about things like IR reflection in instruments in space because they have no atmosphere up in space to keep things cool and stop infrared from bouncing around forever. As a rule of thumb, in large volume systems filled with inert gas like air with no extraordinarily hot things in it (~1000K is where things start to get "extraordinary"), convection tends to be the main mechanism of heat transfer. At smaller volumes and for fluids with higher Prandtl number (like oil and such), conduction plays a big part too. IR radiation is the weakest of the bunch.
But as far as practical advice, I would say do what's been proven to work and don't change nothin' unless you're being paid to do R&D.
