As with any dog, it is virtually impossible to give a straightforward yes or no answer. A lot depends on the dog obviously but also on you, your family and your DC. Some greyhounds live very happily with small furries but they are in the minority as many will chase, this after all is what they are bred and trained to do. A broad rule of thumb is that around 20% of greys are cat/furry safe - talk to a reputable specialist greyhound rescue and they can advise. Be prepared to wait for this sort of dog - there are often waiting lists for the cat-friendly.
WRT DC - 4, 5 and 11 year old would probably be OK but they would need to be absolutely clear that dog is NOT to be disturbed when sleeping, dog is to be given safe place to sleep and to instructed on things like the fact that greys can sleep with their eyes open. Greys are usually very gentle, affectionate dogs, often quite sensitive. If your children are very boisterous and noisy and are looking for a dog that can play in lots of rough and tumble etc a grey is probably not the dog for your family. If however you want an adorable love bucket who will absorb vast quantities of cuddles and take over your sofa then you will be in business. You should also check your DC's position re allergies - have they been exposed to dogs before? There are several posters on here who combine DC with a grund - have a look at the Pointy thread for a better idea.
With regard to expense, you should get pet insurance. You should also budget for worming, vaccination, dental work (for greyhounds this is very common), routine lumps, bumps, cuts and scrapes, a decent sighthound collar and a couple of coats, bowls, lead (NOT an extending lead - this can break their neck), dog bed and appropriate safety arrangements for travellling in the car such as dog guard, safety harness, cage etc. Personally I think it's advisable to have access to emergency funds such as a credit card as even if your dog is insured, you sometimes need to pay at least some of the bill upfront, and particuarly for emergency or out of hours care. You should also take your grund to obedience classes - for a KC Bronze class think about £40 for up to six/eight lessons which will give you the basics and is great fun. Depending on your working/holiday arrangements you should also budget for kennelling when you go away on holiday, or home boarding. If you do go back to work for three days a week then you should use a professional dog walker or make similar arrangements - this too will cost money.
Food costs can vary considerably depending on the type of diet you choose. A good reputable kibble such as James Wellbeloved will be fine, also allow for treats such as pigs ears, the odd dog biscuit etc. Many people, including us, feed raw but this is very much a matter of choice. You can get kibble from the big pet sheds such as Pets at Home, from farm/country stores or if you have a wholesaler near you may be worth borrowing a card.
Most greyhound rescues hold regular walks, or meet and greets at places like pet food stores, where you can go along, meet a few hounds, take the children, ask lots of questions and find out more in a relaxed no obligation way. Our local charity does a monthly walk at a nearby country park and it's great to see about 20 - 30 assorted pointies all strolling along looking gorgeous. Great too for prospective owners who can come along, walk a dog, and get chatting to owners.
If you would like a recommendation for a reputable greyhound charity just say where in the UK (approximately) you are - can then point you at one. Specialist charity probably better as they understand some of grey's more unique features 
particularly on issues like recall and dealing with small furries. Greys are generally extremely healthy, since they are bred to work not for looks and can live to mid teens. They have some unique veterinary features though (again, a specialist rescue can advise) such as higher oxygen content in blood, sensitivity to anaesthetics and slightly different blood chemistry - but that's stuff you don't need to worry about at this stage.
I will be blunt about the recall issue. Any body who owns a sighthound (grey, lurcher, saluki, etc) has to be aware that they are bred and trained to chase. Even if you go to obedience classes and your dog is generally pretty good, if they see something that arouses their chase instinct they can and will run. Very, very fast indeed. You will not catch them. For this reason, you should always be extremely careful where and when you let them off lead. For this reason too, many greyhound charities have offlead secure facilities (such as a special field or often a riding school) where pointies can play safely. If you are not prepared for this kind of vigilance, then a pointy is probably not for you. I'm sorry if this sounds a bit harsh - i don't mean it to - it really is for the dog's own protection.
Happy to answer any more pointy related questions (we have three). They are very addictive