Basic sensor cleaning is really easy. If it really is sticky pollen grains you may need to do (or pay for) a full wet sensor clean, but usually a good blow with a rocket blower (
some examples from amazon) will do the job. Camera manuals are always full of dire warnings about damaging the sensor, but as long as you're sensible there's little risk. Just make sure the battery is well charged, take the lens off and find the menu option for sensor cleaning which will lock the mirror out of the way. Hold the camera upside down and blast the sensor with air a few times.
As to why you're picking up dust/pollen without changing the lens: Yes, the lens-to-body join won't be sealed, although I wouldn't expect much dust to get in through there in normal use. Even though it's not a zoom, does your lens extend/contract as you focus? If so, that can act as a pump to suck air and dust into the camera.