Is dust on the sensor normal? Unfortunately, yes. One of the few disadvantages of digital vs. film is that the recording medium isn't changed every time we swap out a roll of film. Over time dust will find its way onto your sensor. There are several precautions you can take to minimize the impact of dust on your sensor.

1. Change lenses as rarely as possible, and in the cleanest environment that is feasible. If you must change lenses out in the field, place your back to the wind and hold your camera at an angle down and away from the wind. Blow out the back of the lens before placing it on your camera. Run the automatic sensor cleaning cycle each time you change lenses and before and after every shoot. De-energize the sensor by turning the camera off whenever the body is open.
2. Keep your camera in as dry an environment as you can. When not in use, store it with a desiccant. Avoid rapid temperature changes, especially from cold to warm. Stow the camera in the bag in the cold outside and then leave the bag shut for an hour or two when you move indoors. In the summer take the camera outside in the bag and leave it for a while before removing it. Moisture can turn an easy to blow off speck of dust into an almost impossible to remove spot of dust.
3. When you do have dust that the automatic sensor cleaning won't dislodge after a couple of cycles, clean in the following sequence until the dust no longer appears in a test photo. Blower → Brush → Liquid → Tape (if you dare). Each method can remove progressively more difficult dust, but also carries progressively greater risk of damaging the IR filter that covers your sensor. Each method has a learning curve before you can maximize its effectiveness. For example, I still struggle with leaving dust in the extreme corners of the sensor when I use a swab with liquid.

Once you have taken images that have dust spots, you will need to edit them out in some way.

1. Follow your camera's instructions to take a Dust Delete Data picture. For some cameras you need to do this before you take any pictures you want to apply the data to. Canon cameras, for example, append the current Dust Delete Data saved in the camera to the meta-data for each file as you shoot, then gives you the option to apply it using Digital Photo Professional in post processing.
2. Depending on your workflow, you can do dust removal on each image before combining them for an HDR image. If you are manually using a clone/repair tool, temporarily turn the contrast and sharpening up and the exposure/brightness down while hunting for and repairing dust spots. This will help you see them better. Once you've cleaned the dust spots, adjust the other parameters to your liking and export the image as a TIFF file to use in your preferred HDR software. I've found this much more effective than trying to remove dust spots from a finished HDR image.