
Ive got two lurchers here - one ive had from a pup, one from 9 yaers old who is an ex worker (and was im reliably informed, used to hunt CATS!!!

)
Both are fine with MY cat, even outside of the house. However neither are safe with unknown cats(but then neither of my non lurchers are safe with unknown cats either!)
As a pup, they are no more likely to pose a problem in socialising to other dogs and cats or other small furrys than any other breed, but you do have to put the work in.
Knowing how the lurcher is bred often helps with training, collie bred lurchers will still show some herding and driving behaviour, terrier bred lurchers will often show keen terriery behaviour.
Lurchers SHOULD be taught how to play nicely with other dogs, theiri play frequently involves a 'dumbed down' version of hunting, ie chase, grab/trip, pin down..... its rare that they mean business however it can be deeply unpleasant for other dogs so a good solid 'leave' is wise and preferably lots of play with the sort of confident dog that just ignores this lurchery stupidness and says 'sod off you pointy lunatic', as opposed to submissive dogs who will roll over and cry for mummy or really aggro dogs who will fight back.
That said, all dogs need to nkow how to play nicely!
Em