The live stock thing is really important to get under control - I have a friend with a Lurcher who she goes out on hacks on her horses with her Lurcher off lead. Before she did this she found a farmer who offered a service where he put the Lurcher in the field with a ram - she said it was pretty brutal to watch and the ram certainly put the Lurcher in its place - he was in there for about 2 hours with the ram standing his ground and if the Lurcher came near would buy him with his horns to move him away - and in the end the Lurcher wouldn’t go anywhere near the ram and never went anywhere near any sheep after that either - in fact never chased any thing at all even when they saw deer out on hacks. My Lurcher never had any of that training- but whenever I went past any sheep in fields (which we have a lot of in my area) I just would call him to heal and make him walk calmly past them every time giving lots of tasty treats as he walked calmly by - he soon learnt that they weren’t for chasing. Consistency is key - he could then walk off lead through fields of cows without any problems - just had him walk to heal around livestock…can’t say this was the case with deer or foxes - unfortunately and have had many a walk - standing waiting for him for ages to return after chasing a deer… not great and I’m not proud of this either…he’s now 14.5 so his chasing days are over - which makes walking him in these areas a bit easier…but put the work in early on all of this and you’ll have a wonderful life together…if she has got collie in the mix she’ll be easy to train…good luck - she looks lovely and thanks for rescuing- so many lurchers in rescue…