Hi, I currently have two dogs - used to be three - and my eldest has SEN. Will admit to being heavily biased, but have to agree Lurchers are a good candidate, as they are - generally very cakm in the house.
Are you wanting a pup? I guess so from your question about toilet training. ELGR have 17 Lurcher pups, born in rescue at the moment and Lurcher Link have just had a litter of 5 born, so roughly 8 weeks till they'd be ready for their new homes.
Right, toilet training, really depends on the dog, anything fron a week to a couplecof months, but if you search MN and use the method outlined by Lilcamper it should be just a couple of weeks.
If you do get a pup they will need to be taken out and about everywhere, being carried until they compleye their vaccinations, in oder to ensure they get proper socialisation to as many things, people, dogs, other animals etc as possible. The window of oppotunity for this is quite short and you need to get as much in as possible prior to 12 weeks of age.
You will need to get pup used to being alone for short periods right from the off. They can't be left much when they are tiny, but if done right you can build up the length of time they're happy to be left for fairly quickly. Prob no more than a couple if hours for an older pup, but adults can go for a max 3-4 hours. Any longer than that and you'd need a dogwalker or someone to cime in and take them out for you. If we are going somewhere for a long day out, eg themepark, I tend to book mine into kennels as I know they are happy and well looked after there, but there are also dig sitters dig daycare ti consider.
Cost, remember to budget for monthly food bill, training classes, annual vaccinations, including kennel cough if you plan on using daycare or kennels, microchip, regular worming, ad hoc minor vet bills for those things that fall under insurance excess and definitey lifetime civer pet insurance.
Initial cost would be purchase price if pedigree - anything from several hundred to over a thousand depending on breed, or a couple of hundred-ish donation for a rescue. Then dog bed, collar, lead and id tag and toys and possibly a crate uf you decide to use one.
I would recommend you read Life Skills For Puppies if you do want a pup.
Also do bear in mind nipping, super lively pups don't always go down well with some children who have SEN, so well worth seeing if you can vusit a few first and see how your dc copes.
Apologies for any typos, am on phone.