Britrail pass or individual tickets ?

The Elizabeth Line is National Rail, not a Tube Line, so BritRail is valid throughout. Britrail is also valid on Heathrow Express, so no Oyster card is needed.
For Bath to Edinburgh take a train to Bristol Temple Meads, then a train through from Bristol Temple Meads to Edinburgh.
Don't reserve seats through ACT as they charge for what is a free product. They are advisable and can be made at any staffed UK station for any reserveable journey.
If instead you choose to go to Edinburgh from Bath via London then use your Britrail on Elizabeth line Paddington to Farringdon, changing to Thameslink at Farringdon to St Pancras- no Oyster needed.
But with Britrail the Bristol route is better as you aren't thinking about cost and advance fares
This is an unusual case, where for ease of use, Britrail may well justify itself as from Heathrow you can take the first train out, likewise from Paddington to Bath.
No worries about which train is best, or advance tickets on specific trains or what route on what ticket. Likewise the high cost of the Edinburgh direct train (which is one reason why many people route through London) is irrelevant. For you one change and that's it done.

That railmap that you have shown is dreadful. Here is a proper rail map of Britain in diagrammatic form.https://www.nationalrail.co.uk/Rail%20map%20v37%20orange%20Dec%2022.pdf

You can find the schedules on this site:>https://www.nationalrail.co.uk

Britrail can work out more expensive than buying actual tickets in some cases - though is does offer flexibility.

If you go from Bath to Edinburgh, I would go first to Bristol and change at Birmingham and then go up the shorter/quicker West Coast route via Carlisle to Edinburgh. That would allow you to returner back to London (Kings Cross) via the East Coast route. On the east coast route, York is well worth a stop over.

do we have to make reservations with britrail pass ahead of time ? if yes how ? we will mostly be traveling in weekdays

no you don't have to make a reservation - the point of the pass is that it is flexible so you travel when you want and simply find an unoccupied seat. If you want a reserved seat you can do this at a station the day before you travel. You can do them all in one go.

I am assuming we need an Oyster card to travel from LHR to say Paddington station (where we plan to take the train to Bath). is that correct

no, you can use a contactless bank card or cash

Whilst you won't need Oyster for the trip to Bath you will need some form of payment when you get back to London to use the tube and buses. But even this does not need an Oyster card these days. You can just use a contactless card, but bear in mind that everyone ion your party needs their own method of payment - you can't use one card for more than one person. But you can also use phone or smart watch pay as well as actual cards.
Some people do prefer to have Oyster cards as they don't want to keep bringing out a credit or debit card for every fare. Bear in mind that the cards cost a non-refundable £7 fee but the charges will be exactly the same as using contactless, so the latter is cheaper.

If you decide to change at New Street Birmingham for the Avanti West Coast vis Carlisle and Glasgow to Edinburgh be sure that the line from Carlisle is in fact open. There have been problems there.

Likewise it is unlikely that the long term line closure south of Oxford would affect you. but the joys of a Brit Rail Pass are that you can change your mind on a whim and you say, hey, let's check out Oxford and possibly run into a problem.

Both issues above are due to damaged infrastructure.

The West Coast route re-opened today on schedule, at least on weekdays, although weekend closures continue until June. See this thread-
https://community.ricksteves.com/travel-forum/england/avanti-west-coast-trains-to-glasgow-engineering-work-in-scotland

The work there has been to rebuild Carstairs Junction- where the Edinburgh and Glasgow routes diverge. This was long planned work to simplify the trackwork and improve line speed, work apparently deferred from 20 years ago, when the WCML was modernised.
It was not connected with when the line was washed out in the winter floods- damage which was very swiftly repaired.

I'm not sure that I would risk a 13 minute connection at Birmingham, unless my in-bound Cross Country was right on time, when I'm already on a train that will take me to my destination, arriving 45 minutes later.
Tomorrow I have a 2 hour connection at Crewe, deliberately long as I have split ticketing on Crewe, done to travel on a record low priced fare from Cumbria to London of £14.85 for 340 miles of travel. A long connection to cover every likely issue of a cancellation on my local line followed by an Avanti cancellation. If I'm on time I have a side trip planned, but unbooked. I then still have a 5 hour+ connection across London to St Pancras.

You don't need to reserve but can - use Seatrez

You can use Britrail on the Elizabeth line but have to have it validated at a ticket office first

Thanks all for the very helpful comments.
I am thinking of buying BRITRAIL FLEXIPASS M-PASS 3 day pass from https://shop.acprail.com/#/pass/britrail_passes/results/
That should cover our travel from LHR --> Bath, Bath --> Edinburgh and then Ediburgh --> London

we have 3 days of travel (non-consecutive). I think I have the right pass chosen.

since we are traveling with kids, I am considering first class. first class seems to be 50% more expensive than normal fare. worth it ? is it that comfortable ?

First Class is more expensive. Many of the folks in there are business people getting work done, fares paid by their employer.

Not all, but many.

Don't know your kiddoes. Will they be happy to quietly look out the windows or read a book, or put on earbuds and enjoy the device? Or will they perhaps attract "looks" from the other denizens? Will they be OK with that - and will you?