No, your oyster card is not valid for your journey. Oyster validity in the Brighton direction only goes as far as Gatwick.
If you have a travelcard on your oyster card you can buy a rail ticket from the last station where your travelcard is valid or buy a "boundary zone" rail ticket and travel through on a single train.
Otherwise you would need to ...
I assume you mean the Brighton which is on the south coast of England.
If so, no.
You can start the travel on your Oyster Card, but when you reach the end of the validity area you will have to leave the train and buy a new ticket.
It will be easier to buy a ticket from your starting station all the way to Brighton, which will be easier and likely be cheaper.
A coach has a reasonable likelihood of being cheaper than a train, though a train will probably be the more pleasant option. Our roads are often congested so a train will usually be faster, and you'll have more space to move about, larger toilets, a place to buy snacks and light meals, etc.
London Underground is quite simple. If you're using a paper ticket ...
National Express tickets may be cheaper if you book in advance. Buying in advance guarantees you a seat on the bus, if you turn up on the day it might be sold out (that might be less likely at the current time).
London underground tickets aren't bound to a specific train. Get there, buy your ticket, and take the first train towards your destination.
For a flight arriving at Heathrow nominally at 7pm, it is not possible to take public transport to Leeds that night.
Your only three options are
Have (for example) relatives from Leeds pick you up in a car. I'd say £150 would be appropriate for this. (Likely your cheapest option; recall petrol is very expensive in the UK btw.)
Stay in a hotel, probably at ...
what destination should I book? I don't see Leeds Tempest Road in the National Express site.
National Express has only one stop in Leeds, which is the Dyer Street Coach station. From there you need to take other transportation (bus or taxi)
Will we have enough space to keep the luggage in National Express?
The bus has plenty of storage. What exactly you ...