England/Ireland

I guess it all depend on the primary focus of your trip and how extensive your itinerary in Ireland shall be. At first blush, I see nothing wrong with stopping over in London and doing an open-jaw between Shannon and Dublin. Both Dover and Stonehenge can be done as daytrips out of London, so chew 5 days min for that module, if you want enough time to visit London too: arrival day + 2 (for London) + 2 daytrips. You'll need to keep a min 2 days for Dublin at the end. What is left of your days would then be your actual available days to tour Ireland. So, it comes down to the question of how ambitious are your plans for Ireland?

Do you mean to rent a car in England and take it to Ireland? That may be a little difficult. You won't want a car in London and certainly won't want to pay the London Congestion Charge. What do you want to see in Dover?

You can spend 14 days is southern England and not be bored. You can certainly spend 14 days in Ireland and not see all the highlights. You are posting a variation of "can I see it all" which appears here frequently. My answer is that you need to put together a detailed plan for your trip and decide what are the most important things to see. I personally think Stonehenge is over-rated but many here will disagree. Avebury was more fun and there is a National Trust property there that my wife and I both enjoyed. Wells was awarded my personal "best" for evensongs. Clonmacnoise was definitely worth the stop driving between Dublin and Galway. Yep, 14 days barely touches either England or Ireland and I haven't even mentioned Scotland or Wales.

The answer is yes. Logistically consider flying into Heathrow, see London, then off to Stonehenge and Dover on day trips. Each can be reached by train or coach (bus) travel. No need to rent a car. From London fly to Dublin and tour Ireland, then fly home to the So. Bay Area from Shannon. Eire is where having a car is a must. Use Google maps and you'll see travel time is only 4.5 hours from Dublin to Shannon on their main freeway. Shorter than the 6 hour trek from Cupertino to LA on I-5 and a far nicer jaunt. BTW is my favorite falafel spot Vivi's still open on Steven's Creek.