You need to look at those with a red tractor logo - those are working farms. You may or may not be allowed to view the animals though, and staying on a farm could make it a pain returning to the US. Find somewhere you like the look of and contact the owners to see if you can visit the farm animals. Unless there is a reason to bring the sheep in, shearing, or whatever you are unlikely to see dogs in action. You would be better off searching for sheep-dog trials going on while you are visiting.

Self-catering - you do as much or a little cooking as you want, there is nothing to stop you eating at a local pub for instance.

"Also, I don't know what "self-catered" means, but it sounds as if I'd be cooking. Is that right? "

Self-catering is just Brit-speak for 'vacation rental'. The same as renting a condo or cottage in the States. Cook if you want, don't cook if you don't want.

You will see LOTS of animals w/o having to stay on a working farm. As long as you are in the country you will see sheep, cattle, horses . . . they are everywhere. Staying on a farm is fine - but don't choose one simply so you can see animals. And as mentioned, many working farms won't want you mucking about w/ the operations.

Instead - look for village fetes in the areas you are staying . . . There will often be sheepdogs, sheep, etc. Ask at the tourist info office for local fetes.

Be aware that returning to the US you have to inform Customs if you have spend time on a farm. Concerned about hoof & mouth disease - not sure what they do. Sanitize your shoes?????

You can visit a city farm in London:
http://www.timeout.com/london/things-to-do/city-farms-in-london-1

You could combine a visit to Mudchute City Farm with a trip to Greenwich, if that's in your itinerary:
http://www.mudchute.org/

How near York do you want to stay? This place is a fair trip away but sounds right up your street maybe for a night, and spend 2 nights in York itself?
http://www.bondcroft.yorks.net/page1.asp

Apologies - it doesn't take children!

Our family spent a week in this cottage on a large estate (Birstwith Hall) near Harrogate as part of an extended trip to Europe a few years ago: http://www.yorkshirenet.co.uk/stayat/dovecote/index.htm. There were horses stabled off the courtyard next to our house, and our daughter really enjoyed helping care for them. There weren't other farm animals at that time, but as janisj says, there are farm animals everywhere in the countryside. When you hike on public footpaths in the UK, you'll walk right through fields of sheep, cows, sometimes horses.

We loved this part of England...

Kathy