(I feel like I'm taking the College Boards or something!)
First of all, Dexters are considered a dual-purpose breed, milk and beef in quantities for a family (some train them as oxen too), small acreage cattle, easier on pasture space, feed requirements, etc. They are small, but not "miniature" (although a lot of people who breed "miniature cattle" use Dexters). A few want to have something "interesting" in their pasture, a pet...please don't faint, Ollie.
We've had them for 6 years...one year, when we had five bull calves, I 'freaked' wondering what I'd do with all those boys...but a number of people called and wanted their own personal beef animal to raise, so we castrated them and sold them. Finally got around to having one processed for ourselves last fall, because I did not want to sell what I had not used myself (and now I'm afraid I'll run out of beef before the next one's ready). Most people want a heifer for breeding or a family milk cow...at weaning, we halter train the heifers and handle them (brush, tie, lead, etc.) and it's a great feeling to lead one out of the pen, across the yard and into the trailer (even if she's never been in a trailer before). They're vaccinated and registered before they leave (we don't register the steers).
Improving stock? Yes, that's our hope...we got a new young bull last fall, unrelated (as much as possible) to our cows. We also have semen from a Canadian bull and a British bull. Trying to broaden our own genetic base here. Being a rare breed (though now officially upgraded to "watch" status by the Amer. Livestock Breeds Conservancy), the Dexter gene pool could be considered small. We had our first calf by the British bull in January.
I would say that Dexters are in a transition stage at this point in time...red is on the increase, as are polled Dexters (lots of controversy there!). Our animals have horns and a lot of people don't want this; we feel it's okay to de-horn a young calf but not an adult animal. Personally, I think the horns are part of what makes them Dexters, but I understand that not everyone feels that way. Also, while some breeders are gung-ho on red, polled animals, other breeders are attempting to preserve the "old" genetics. Interesting times at the dexter corral, to say the least....
Responsibility? Yup, not just in breeding responsibly to avoid some genetic problems (chondrodysplasia "bull-dog" gene) but also responsible breeding (age factor) and responsible care (feed, hay). Dexters' reputation as easy calvers, hardy, good foragers, etc., would lead some people to believe they can be exposed to a bull at 6-7 months, and can eat tree-bark to survive. There are some breeders who do not routinely vaccinate their Dexters at all.
I could go on and on, but hope I've answered your basic questions. Dexters aren't for everybody, but we like them, are interested in improving what we offer. I'd like to have a milk cow here too, but just don't have the time to do that yet...
Gale