Paint the door to match the white trim.

I would suggest looking through www.flor.com. They have some great, very durable, products that would add some pizzazz to your entryway.

I respectfully disagree with the previous poster...I love your doors!
As for the question at hand, I would recommend an indoor/outdoor rug that you can literally hose off or something made with natural fibers like wool that you can spot clean. I would also suggest something with a pattern as it will hide spots and hair better between cleanings.
Overstock has a pretty decent selection of both kinds of rugs.

Surya has some great options and prices for area rugs..Polypropelene rugs are very easy to clean up and bleach will not discolor them but just do not have the longest life span but the price on them is great and change styles often without breaking the bank like on a good wool rug. Wool rugs are very nice but shedding dog and shedding wool rug ehh not so much.