Do UK and Ireland each have separate visas/regulations?

What is your citizenship? If it is USA then you do not need a visa for either the UK or Republic of Ireland. If you do need a visa then yes they are separate countries and require separate visas although the recent British-Irish Visa Scheme is starting to provide commonality, starting with visitors of Chinese or Indian nationality.

If you are really interested in periods of stay rather than visas per se you would in general be allowed 6 months stay in the UK on entry through Dover (presuming ferry) or in Paris (assuming Eurostar train). You can stay for 3 months within the Republic of Ireland.

If you travel by ferry or www.eurotunnel.com from France to Britain, I'm sure your passport will be stamped. If you are leasing a car in France, such as through www.renaultusa.com, then you make a specific appointment to return the car at a certain time and location with an agent who meets you there, and you pay a fee to leave it outside of France. I found it no problem to drive in Britain and Ireland with such a leased French car, except if you come to any turnstile or toll booth and don't have someone seated on the right side of the car.

Yes, UK and Ireland each have separate visas/regulations, but there are no passport controls between the two countries (they have their own "common travel area")
Yes, there passport checks between Schengen countries and the UK / Ireland.
As a non-EU citizen you will have your passport stamped as leaving Schengen-area when crossing the Channel, and stamped again for entering the UK. On the return journey the British DO NOT stamp on exit, but Schengen will stamp on entry. This is your proof you have not exceeded the Schengen 90 days in 180 rule. This is only for non-EU/EEA citizens. EU/EEA citizens have no time limits and do not have their passports stamped.

UK and Ireland have separate and independent visa rules. So long as you keep under the limits for each country you will be OK.

Ferry from France to the UK is slow and inconvenient. The ferries offer a cheap and slow alternative for car travellers trying to save money, Use the car shuttles through the Channel Tunnel instead.
But, check with your car hire company that you are allowed to take your (left-hand-drive) car into the UK. They think it as additional insurance risk to drive a car with the steering wheel "on the wrong side" in the UK.

A good if not best staring point for answers to visa and foreign travel questions is the US State Department if you are a citizen.

http://travel.state.gov/content/passports/english/country/ireland.html

http://travel.state.gov/content/passports/english/country/united-kingdom.html

If you are driving in Ireland you may wanted too review country specific State Department advice:

More information on driving in Ireland can be found on U.S. Embassy
Dublin’s website. For specific information concerning Irish driving
permits, vehicle inspection, road tax and mandatory insurance, please
visit the official tourism guide for Ireland.

http://dublin.usembassy.gov/service/other-citizen-services/other-citizen-services/driving.html

It is great to have this forum for those little details that nibble at my brain! Many thanks. I am back on track now.

Yes, I am a US citizen, and while I visited the State Dept website I somehow managed to miss the very obviously stated max stay in Ireland, 3 months, and UK, 6 months. I am not sure how they would know how many days I stayed in each respective country if there are no checks when moving between, but that was only a curiosity on my part for future reference.

Thanks for the passport control details. I assumed something like that would happen but now I know for sure.
Travel by ferry would be a choice simply because it would be different, and I like being on the water.

As for driving on the left - that will be a first for me, and I intend to stay away from congestion as best I can until I'm comfortable. At least on the left I will know how close to the edge I can get. Toll booths etc are a problem but perhaps mitigated by using an extended "gripper" to pass cash, or a small basket on a stick. Any other ideas?

Thanks again. j2

You are required to have a passport to travel on ferries between the UK and Ireland. British and Irish nationals though need only carry something that identifies them and this can be as simple as an electricity bill! The UK / Republic of Ireland border used to be completely unmarked though in recent years there have been attempts to put up 'Welcome to Northern Ireland' signs on some main routes.