What should I use to fill cracks in my wood door?
Word of advice, do not sand the door anymore than what has occured already. You can see how there are a few lighter areas where some dings are that the "patina" has been removed by something. Sanding will ruin the patina, making the door look blotchy as well.

On old furniture and doors where the cracks are not structural, burn in sticks are what I use. They are made of lacquer, and if you finish your door in a water based or poly finish, this will fill the cracks nicely. Burn in sticks can be bought on Ebay, its where I got mine, they come in a wide variety of colors, and the color you see is the color you get, they don't change color once applied.

Once the door is sealed, NOT finish coated you get yourself handy a heat gun. Use that to melt the stick of the color you want where it starts to flow down the stick, careful. Then move it onto the door with out dripping and push it into the crack. On small cracks, I use a putty knife heated and set the stick on it to melt it a little and use the knife to apply it. When the crack or hole is sufficiently filled, I take the heated putty knike and flatten it out, removing the excess. This will leave a small skim over the surface, which will then clean up with a little lacquer thinner on a clean rag. Do not get the rag to wet, it may try to dissolve the sealer too. 

Apply the finish coats over the repairs as you normally would.

Use this only in cracks, not joints, wood needs to move and if it is placed in a joint you will hinder the movement, doing more harm than good.