Need help with train tickets and passes. What do I need or what should I get?

Yes, book your journeys outside London in advance – look for Advance tickets on sale about 12 weeks before you travel.

With a Friends and Family Railcard you’ll get one-third off the prices. Buy your tickets online indicating you have the railcard. When you're in the UK buy the railcard before your first journey.

The Family and Friends Rail Card should certainly be a benefit, giving the adults about 30% off and kids about 60% off the regular range of prices, whether or not you book in advance for additional savings. You have to decide how important and likely it is that you will have the whole party ready in time to board a particular train. If your research has shown that York to Edinburgh, for instance, is the most expensive segment, then it may be the one you're most inclined to book ahead. On the other hand, with trains running so frequently, you may prefer a flexible schedule.

For comparison, if you choose a BritRail Pass to cover the 4 longest trips listed above (within a month), then the price is $278 per adult and kids are free. Seat reservations are not required but are free to make up to a day ahead in train stations, if you want assurance of seats together.

Yes use the Oyster in London and given the age of your children ask about the 5-10 zip Oyster.

Might take ID photos of your kids before you go and have them if asked for them. Hopefully some one on the forum who has used a zip Oyster for their children can explain how to obtain .
https://www.visitlondon.com/traveller-information/getting-around-london/do-kids-need-an-oyster-card

Yes. Battersea Park Station is a rail line ( Southern) where you can buy rail tickets . It is also an Overground Station where u can use your Oyster card. You can use the Oyster on Southern trains.
https://www.southernrailway.com/tickets/smartcards-oyster-and-plusbus/oyster

Pretty sure the "zip oyster" is really meant for children that live in London and need to travel on their own (to school etc), the "young person discount" can be added to a normal oyster card but that's for 11 to 15 year olds, under 11's travel free with a fare paying adult.

My trip is much simpler but for reference I just booked 5 adult one-way tickets from London to Edinburgh for 126 pounds. There were the "Advance" tickets and I used the LNER website. I am about 12 weeks out.

Children under 11 travel free if accompanied on the Underground, Overground and DLR, and unaccompanied on the bus. Not free though on national rail services.

An F&F card does give a 1/3 discount on one day travel cards for the adults, but there probably are cheaper ways, unless you want to use those for the Two4One admission offers.

I'm not certsin because I'm usually a solo traveler, but I think the Family and Friends railcard requires that everyone travel together, which might be a probkllem if you wanted to separate for a day because of different interests or becsuse someone was ill and not up to traveling. Read the details before making that purchase.

F&F card is valid with just one named adult plus one child using it.

There is an uncertainty with the digital version whether each named adult can travel on different journeys with a child at the same time with the card held on different devices.