That will depend on your own nationality and where you are initially travelling from. Contact your department of foreign affairs, or the Irish one for more information.
definitely, yes
yes. which ever way you leave a country, u need a passport
no, as long as you have a passport
No. They are both in Ireland, so if you are in Cork legally, then going to Dublin does not require any additional documentation. You may have needed a passport to get to Ireland in the first place. If so, then keep it with you when travelling around Ireland.
no you dont
You need a passport to enter any country, that's why you have passport control at airports.
Yes you would need a passport. You would also need a visa.
As the "Common Travel Area" is in place, you do not need a passport to go to Dublin from the UK. When flying between the UK and Ireland it is best to ask your airline what form of ID will be accepted.
Not unless you need a passport to visit Weymouth.
Yes, you will need to show your passport.
You need a passport to go anywhere.
No, but you will need a passport to get back into the US.