The fan noise is the cooling fan for the radiator; in the ambient temps you describe, that's about right for the coolant temp to hit the point where the fan will be needed to keep the coolant at a steady temperature.
My guess is that the slight push you're feeling is probably the compressor cycling in the climate control. Are you running the climate control on Automatic? If so, the compressor will cycle and when it does, it has the effect of slightly increasing engine rpm, which will be noticeable if you are stopped with light pressure on the brake pedal.
2014 GLK250 Diesel
History: '69 220/8 [OSD]; '71 220/8; '73 SE 4.5; '76 300D; '77 240D; '78 240D; '79 300SD; '81 300SD; '84 190E; '86 190E; '89 260E; '94 C220; '98 E320; '02 C240; '06 C280; '08 C300; '10 GLK350: '14 GLK250 BT