They should have waited one more year and then done the landing at the Pas de Calais, the extra preparation would have overcome the stronger German defenses. Do some more stuff in peripheral areas while preparing. Pretty much the British idea.
1. Could the allies gained a sufficient foothold to mount a full-on assault on Europe?
There was a big problem with the Normandy site. While the German defenses were supposed to be weaker there relative to the Pas de Calais (and that doesn't even seem to have been the case, one instance where Hitler's intuition really worked), the ports were inadequate and even if the Allies had overcome that, they had destroyed the French rail network so they would have run into limitations on supplying their armies inland. If the landings had worked and not been contained, the smart thing for the Germans to do would have been to just run away to a new defensive line much further inland and it would have taken the Allies time to catch up and get their logistics caught up.
2. How much sooner would the war have ended in Europe?
Despite what I said earlier, a lot sooner, just given that the Germans now have to mix it up with the British, Americans, Canadians, and whatever the liberated French can put together, and outside of the Italian peninsula where two armies and defensive lines can contain them. Maybe even in 1944 if there is a complete German collapse, for instance they hold on too long in Normandy and get their western armies destroyed. There is also a big potential butterfly you are ignoring with Operation Valkyrie, a lot of the principals are now completely differently positioned.
3. Would the Allies still have used the A-bomb on Germany?
No, if they can get a German surrender before July 1945. Or even if the Germans are really obviously losing by that point.
4. And what of Eisenhower - what would have been the trajectory of his career if his plan had actually worked?
I wonder if the US tradition of of making successful generals President would have still be operative, or if that was just a nineteenth century/ American Civil War thing. Eisenhower may not even have been interested. Despite a lot of ballyhoo and a much bigger ego, MacArthur never actually entered electoral politics. Eisenhower could have just retired and wrote his memoirs like IOTL with a higher reputation (the court martial was a pure PR exercise and Eisenhower was never in serious danger of even losing rank).