Hi!
You can apply through either a voluntary agency or your Local Authority, and depending on where you live maybe the neighbouring Local Authorities. There's pros and cons to each.
It doesn't cost anything unless you go international, in which case you will be charged a few thousand for homestudy. There are court costs at finalization but the agency/authority should pay for you. Then you get to sit back and watch the large child shaped hole in your bank account grow...and grow!! 
Very different with different people. This depends on several things. Personally, 1 and a half years.
But it depends. What age, gender and number of children are you hoping to adopt? Are you willing to consider certain disabilities. How much can you handle? All very important. If you are approved for a older child, say 7-10, boy or girl, you may get an overwhelming amount of interest. For a girl aged 0-18 months, you may wait years.
Most children have a form of ongoing contact with their birth families all their childhood, some don't. It may be by letterbox, a system whereby you write to the birth parents/ siblings/ GP's and if you are lucky they write back. Many don't return letters, some are too disorganised, for others it is too painful.
Some children have direct contact, by phone or usually a meeting once or maybe twice a year with their BP's/GP's. They may have more meetings with brothers and sisters.
At adulthood yes, either your child could trace, or their birth family can trace them.
Feel free to post more questions, this is very basic info, I'm happy to help.
Afraid I don't know a lot about other countries, only that the current wait for China is like 7 years!!