Trip to United Kingdom

For tickets to musicals, there are A LOT of walk in ticket-booths around Leicester Square and between there and Piccadilly Circus. They list out all the prices and you can even compare them before you commit. The more established one is called `tkts`; it`s a huge pavillion-like booth right in the middle of Leicester Square. Go early, like before 9am, if not you`re quite likely to be caught in a long queue and your chances of getting cheap tickets would diminish.

Accoms-wise, I can`t advise you... sorry about that.

According to your itinerary, the cheapest you`ll need is the BritRail England Consecutive Pass, costs USD209 for yourself and the boy can travel with you free on your pass. But, if you look around now you might be able to reserve bus tickets online for as little as 1 pound per trip; I have also gotten train tickets for 8 pounds one way from Cardiff to Bath. Check the prices first and then decide whether you should get your rail pass.

It would make the most sense for you to go to Cambridge (day trip - 45min train from King's Cross) while you are in London, stay for a night or 2 in Manchester/Liverpool while you are touring those cities, and visit Newcastle while you`re in Sunderland.

Websites:
BritRail Pass:
http://www.acprailnet.com/britrailbook?returnurl=http://www.acprail.com&countrycode=HK&languagecode=EN&countrykey=ACP09873

Buses:
www.nationalexpress.com
www.megabus.co.uk

Trains:
http://www.nationalrail.co.uk/

for your 8 year-old is FRE, but you will need a photo card for him. Just read the info below from the TFL. have a passport photo will you and your child is free to go in London.
5 to 10 year-olds
Children aged 5-10 years can now travel FREE at any time on the buses, the Tube, trams, DLR and London Overground at any time.
On buses and trams

Unaccompanied 5-10 year-olds can travel free at any time. They do not need a ticket or photocard to do so
On the Tube, DLR and London Overground:

Up to four 5-10 year-olds can travel free when accompanied by an adult who has a valid ticket, is using Oyster to pay as they go or has a Freedom Pass
If they are travelling unaccompanied, 5-10 year-olds need a 5 to 10 Oyster photocard to travel free at any time
Under-11s travelling with an adult will not be issued with tickets and will be allowed to pass through the ticket gates at stations. Please note: if they have an Oyster photocard they should not use it when being accompanied, or they could be charged a fare.
to have good sit, you do need to book and most show have website, and you can book ahead. Opera is not big in UK, but theatre is popular. check out the below link.
http://www.visitlondon.com/search?keyword=opera
if you plan to go out most of the time and top hotel is not what you want. travel lodge and holiday inn, are good choices.
http://www.travelodge.co.uk/
www.holidayinn.co.uk
those hotels locat all round UK.
new castle is about an hour drive, liverpool and Manchester for day 6 and Cambridge on the way back to London.
the link below is a good map to use.
http://www.multimap.com/
you can find alot of info with this link.
http://www.visitlondon.com/
http://www.tfl.gov.uk/tickets/default.aspx

there are a lot of trains. i suggest you to buy a few days pass of metro.