Talk about the use of 'could' and 'would'

Hi FBers, we have been asked to talk about the use of 'could' and 'would'.

When we ask someone to do things we can use either 'could' or 'would':
Would you help me carry this please?
Could you open the door for me?

When we ask for something we use 'could':
Could I have a coffee please?
Could I see those shoes?

When we ask permission we can use 'could' or 'may':
Could I speak to John please?
May I come in?

When we offer something or invite someone we use 'would':
Would you like some tea?
Would he like to come to the beach tomorrow?

When we talk about something being done continually in the past we use 'would':
When she was young she would play on the swing for hours.

When we talk about the ability to do something in the past we use 'could'
As a child he could swim like a fish.