What are ways to reduce drag and noise generated by the landing gear?
The biggest noise reduction is achieved by closing most of the landing gear doors again after extension is complete.

Of the factors that influence drag and noise, one is the distance between wheel and fuselage. The picture below is taken from S. Hoerner's book "Fluid Dynamic Drag", chapter XIII.5 of which deals with the drag of landing gears.



Another influence is the shape of the landing gear strut. Since most airliner landing gear legs are oleo struts, they are round for practical reasons. Fairing them would already help, as can be seen from the lower line in the plot above. Another improvement can be achieved by covering the wheel hubs. However, the reduction of the drag and noise caused by extended, retractable landing gears has been studied but few of the results have yet found their way into actual aircraft. According to this DLR report, a fully faired A320 landing gear would reduce noise by 10 dB while partially fairing the most noisy parts of an A340 landing gear would reduce noise by 5 dB. The picture below is taken from this report.



For fixed landing gears, I recommend NACA Technical Report 485 which contains results of full-scale tests on several gears. NACA Technical Report 518 contains results for partially retractable gears.