How long and how often do you want to walk a dog?
Ideally 2 30 Ishminutes walks on beach each day, and a few shorter walks to garden and park.
Then you are looking for a dog with low - low/moderate exercise needs. This rules out a lot of working breeds, unless you get a particularly lazy one.
How much training and for how long? What type?
Would want to train the dog to be genrally obedient , ie good recall, lead walking. Possibly crate train too depending on dog type I guess.
This all sounds good but is basic training then only. So nothing that had too intelligent/active a mind that requires training or similar for the rest of it's life. More a 'train and then done' type?
How much grooming?
My husband is mildly allergic , so not lots, prob short hair best, so another reason for lab/whippet.
Labs moult like crazy but actually allergies can be triggeree by dander (skin cells) and people can be allergic to one breed but not another. I would stringly recommend your husband spend time around whatever breed you choose BEFORE you commit to one.
How much do you want to spend per month?
Not worried about money, whatever it needs
To what extent do you want the children to participate in exercising and training?
After school and weekends!
I meant more what do you want them to do, e.g. if you want them to walk the dog then it needs to be a size and temperament they can physically hold if it decides to run for something. If you wanted to get them involved with something fun like agility then it needs to be a dog with the capability and desire to do it. etc.
Who will look after the dog when you cannot? What are their limitations?
Friends and family. Very flexible. But both me and husband self employed from gone, so very flexible.
How big is your car?
Berlingo
Ha! No restrictions on space then 
What kind of outdoor space do you have?
10m square garden
That's quite small which can sometimes work in the dog's favour as it means you are more likely to walk it then think the garden is anything other than a toilet. However, with just 2 x 30mins a day, this needs tobe a dog without much energy to burn off, I think.
What dog behaviours can you absolutely not accept? Digging? Chewing? Barking? Guarding? Boisterousness? Shyness? Overly friendly? Etc etc
As in a flat, barking would be difficult. Would prefer an affectionate dog to bond with children.
That would tend to rule out terriers, which can be pretty vocal - though this does vary by breed and individual.
How strong and fit are you?
Average!
No huskies for you then
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All in all, companion breeds might be a nice choice. They tend to be:
- small so easy for children to handle/hold leads etc
- have long socialisation periods so more likely to learn to be friendly with whoever they meet
- not need much exercise but be pretty playful and fun loving
- fit nicely into a small garden
the 'downsides' are...
- some can be sensitive
- there can be a tedancy to treat them like a toy, rather than a dog, pivking them up all the time etc.
- they tend to require a bit of grooming (but being small they fit on your lap to do it)
- they tend to have longer fur so an allergy check swith your husband is a must
- many of the breeds have high amounts of helth issues so you need to take extra, extra care to choose your breeder carefully plus insurance tends to be a bit higher as a result
It can be hard to find a definitive list of companion breeds but look for something that doesn't have a working history, or that has a working history very far back but has since been kept as companions. Someone upthread mentioned a havenese, which is a lovely choice.