Manchester & Liverpool for 3 night

Both Manchester and Liverpool can easily fill 3 days each. They are close enough that it makes sense to stay in one and visit the other, rather than transferring if you only have 3 days in total.
2 days in York and 3 in Edinburgh are fine, although of course more could easily be had.
Longer train rides will be cheaper booked in advance, but then you would have to fix train times - miss them and you would need a new ticket.
You say we - how many in your party? Your travel will be enough to justify buying a railcard but which one depends on the number and ages of you all.

We are party of 3, age 63, 59 and 22. I read about rail pass and need some clarification. Should we buy a student pass and 2-person pass or a senior pass and 2-person pass? Does the pass offer any discount for York to Edinburgh trip since it is out the England area?

In Rick's England tourbook, there is even no mention of Manchester

Has he even heard of anywhere outside of London and the Cotswolds?

We are party of 3, age 63, 59 and 22. I read about rail pass and need some clarification. Should we buy a student pass and 2-person pass or a senior pass and 2-person pass?

Yes. A Senior Railcard and a Two Together Railcard. But be aware of the weekday 9.30am time restriction on the Two Together Railcard

Does the pass offer any discount for York to Edinburgh trip since it is out the England area?

Yes, the railcards do. They are valid throughout the UK- England, Wales and Scotland

In Rick's England tourbook, there is even no mention of Manchester.

This is essentially because the RS guidebooks were originally written (and to an extent still are) as adjuncts to his tour programme. None of the tours route via Manchester (there is only so much ground that can be covered in a 7 or 14 tour) so Manchester doesn't make the cut for the guidebooks.

I spent 2 nights in Chester, 3 nights in Liverpool, and 3 nights in Manchester in the fall. While I enjoyed all three, I’d choose to stay in Liverpool if I only had 3 days. The tour of the Beatles Childhood homes offered by the National Trust was a highlight for me, but I also visited the Western Approaches Museum, the British Music Experience, the Museum of Liverpool, the Beatles Story, and took the very touristy Magical Mystery tour. I saw a musical at Royal Court Theatre and there were lots of interesting restaurants along Bold St.

In Manchester, I saw two concerts at Bridgewater Hall, took a walking tour, visited the Science and Industry Museum, the Art Gallery, the John Rylands Library, and the Central Library.

Chester is also an interesting town for a visit with its medievel shops, original walls, and cathedral.

I don’t think you will run out of things to do in 3 days.

Depending on your interests, Manchester offers a number of things to do.

Here’s an older thread that lists quite a few.

https://community.ricksteves.com/travel-forum/england/a-visit-to-manchester

https://www.visitmanchester.com/

I’ve spent several days there as a departure point for the Lake District and Yorkshire and have enjoyed my time there.

I also spent several days in Liverpool doing several Beatles tours and visiting other museums and sites, and I still didn’t get to do everything I wanted to do.

So between the two cities, you should find more than enough to do during a 3 night stay.

Once your schedule is set, booking the long distance train tickets in advance is a good idea.

All the best,

Raymond

Some more suggestions for Manchester:

Therre is a branch of the Imperial War Museum in Manchester.

The National Football Museum (that's soccer to us colonists) is located near Manchester Arndale.

Machester's public transit system (Bee Network) offers two free bus lines (#1 and #2) that transverse the city center and pass most of the major sights. They originate just outside Piccadilly rail station but can be picked up anywhere on the route. Sort of a free hop on hop off but without commentary.

Warning--they get very crowded with tourists.

Thank everyone for the great suggestions!

Before this 3-days segment, we also have other cities for the trip:

After 4-night in London(1st visit of the city is 4-night 7 years ago, visited Beckenham Palace, British Museum, Royal Observatory Greenwich, Westminster, Tower Bridge, Canary Wharf, Hyde Park),
Plans to visit London Tower, more British Museum, Tate Museum, walk along Tame River,

nor sure we should visit Cambridge since we already planned to visit Oxford for 2-night, taking into consideration we have to travel to King's Cross from our central London hotel

we will take morning train to Slough, check-in Moxy hotel near train station at Slough, then visit Windsor area, overnight at Slough).
Day 2, train to Oxford, visit, overnight Oxford,
Day 3, more Oxford, overnight at Oxford,
Day 4, train to Bristol, visit Bristol, overnight at Bristol
Day 5, day trip to Bath, overnight at Bristol
Day 6, train to Manchester, to continue to our 3-night at Manchester/Liverpool segment in my original post.

Is the above itinerary reasonable?

Do I have to purchase advance tickets for London to Slough trip or trip between Manchester and Liverpool since both are short distance trips? Does railcard cover short distance trips?

I assume that railcard does not cover subway in London, right?

On the Senior railcard only (not the 2T) if you are using Oyster for the London subway, then register your railcard with gateline staff on the Oyster and you get 1/3 off London off peak subway/tube fares.
For Manchester to Liverpool you don't have to buy ahead. However there are cheap Advance train specific fares. They are available until 15 minutes before travel, subject to quota availability.
Yes railcards cover all trips, no matter how short.

I’d say slow down. Both Manchester and Liverpool are worth 3 days each.

The reason we choose Manchester vs Liverpool as a base is that there is direct 3-hour train from Bristol to Manchester no transfer required. We may save Liverpool for next trip if Manchester offers more attractions for 2 and half days, or just take a quick train to Liverpool to visit 3 great buildings at river front.

Thank you for all the comments and suggestions.

The nice thing about those trains is that they start at Bristol Temple Meads- so plenty of time to board.

They are also generally the cheap trains on that route, and the less busy ones, at least as far as Birmingham. Very often the best fares on those trains are found by split ticketing- Bristol to Stafford (not Birmingham New Street as you might expect and Stafford to Manchester. I found that out by trial and error.

I have done Paignton (Devon) to the West Cumbrian Coast for under £30 railcard fare on those Bristol to MAN trains both with a Stafford/Crewe and with a Manchester change- last summer. So a lot further south and a lot further north than you are going.