Sadgurl, greyhounds rarely get sick so you are not likely to need a vet beyond the routine shots/worming etc. If you can try asking a vet for their opinion on greyhounds and their likely health problems. You'll be pleasantly suprised as out of all the pure breeds they are the least likely to develop health problems or catch canine diseases.
As for exercise; greyhounds are sprinters by nature and like human sprinters a quick run every now and then is all they need. They do love the occasional long walk though but they are quite likely to trot along by your side demanding to be stroked as you walk

. However as your neighbour said they are real couch potatos and are more than happy to lie around doing nothing if you'll let them
You really don't need a big enclosed yard for them to burn off energy in like you would with a terrier, lab or a collie. These three breeds are only a few generations from being pure working dogs where they were bred for brains, high energy levels and stamina.
Not suprisingly the dogs brought most often to the rescue center where I worked as vicious "biters" were these breeds with collies being the worst of the lot. Of course they weren't really vicious just bored and frustrated with all their pent up energy.
This is never a problem with greyhounds as they are bred solely to run short distances very quickly. Brains are not required for this and they are so dumb they never get bored

. Lack of brains is a real benefit in dogs living in small spaces because they just don't get bored or curious and it makes them wonderfully placid and affectionate too.
Anyway if I haven't convinced you of the benefits of greyhounds why not think about the other smaller sight hounds? Whippets for example have many of the virtues of greyhounds in a smaller package. The only problems with them is that there is a smaller breeding population so inbreeding can be a risk with all the associated health probelms and they can be a little frail and timid compared to the good natured affectionate nature of greyhounds.
As for mutts you are right to say that they are amongst the healthiest dogs but they often have the same problems living in small spaces as pure bred working breeds ie. too much energy and so smart they get bored and cause problems or eat the furniture.