The F&F railcard is definitely the best plan. I travel a lot up and down the UK with my 2 young DC and have always paid for 1 child's seat as the discount means that the combined price of q adult and 1 child is less than 1 adult alone. I get annoyed on my solo journeys now as I pay more!
If you are travelling scotrail you won't always be able to reserve seats. Virgin or cross country you can.
My first tip would be NOT to book through trainline. First you pay a fee but more importantly if you book directly with the companies you can use their online seat selectors. And if there's a problem you can call and amend by phone which doesn't work if booked via trainline. And please never book the quiet coach!
I have only done it with 2 small kids so always book virgin's coach at disabled/babychange toilet end, and pick one of the 'priority seats' (not the same as the disabled or companion ones) as they have extra leg room. I then sit on the aisle and it blocks the kids in.
However in your situation I would be tempted to either book all of your kids a table seat, or be sneaky and book 3 of the seats (so pay for 3 kids) round a table of 4. It's then unlikely anyone will pick that last seat, or choose to sit in it when they see you.
The buggy provision is rubbish. My maclaren didn't fit either overhead or in the luggage rack (the racks have 3 tiers so it was too long) a slimmer umbrella fold is better as goes above. I have ended up folding and asking the guard. Without fail, they tell me to leave it in the disabled space which I don't like doing.
It really isn't as bad as you think. Pack lots of snacks, quiet toys and don't be afraid to use tablets (with headphones or super quiet). People are great at helping when they see you with kids on your own IME.
Last tip, use the trainline app live arrival bit to plan any changes as you travel, as they show the platforms before they are announced.
Good luck!