It would be an early European theatre war that will be dominated by Prussia and her allies. Prussia had literally dominated Austria and France OTL through the use of railroads and better weaponry. There's no reason to think they wouldn't do it against both. Prussia has faced both at the same time before, and with Italian/Russian support, Austria and France would be facing a serious threat.
England is the unknown-would they jump ship with their biggest rival in Europe now or join their old enemy in France and try to re balance Europe?
1) Prussia dominated Austria through superior weaponry, roads played less a part in that war than simple weapons tech.
2) Prussia dominated France due to their reserves, not their weaponry - in fact, the only place where Prussian weaponry was better than French was when it came to artillery. The French standard rifle in 1870 was better than the Prussian counterpart.
3) Railroads were important for kicking off their opening invasion timetables. After that, not so much other than supplies and reinforcements, and by then France was all but overrun.
4) Prussia never really had superior numbers - put up against both will seriously threaten Prussia's abilities to wage an effective war against both. Perhaps a Schlieffen sort of plan.
5) As mentioned, the 'modern' Prussia had not dealt with both at the same time - and the older Prussia would have gotten their asses handed to them.
British, Italian, and Russian involvement won't change much in the opening stages. If France is allied with Austria, Britain will back Prussia. This reduces France's available manpower as they'll have to watch the sea. Russia and Austria at the time are largely on-par for tactics and tech. That is, they both suck. Russia can overwhelm them though, once they get going... and before Russia falls apart. Actually, Russia at this time might not even be a match for Austria since they perform so terribly in the war against the Turks later on...
I think it is possible and actually rather easy, all you need is France and Austria in slightly better terms than OTL. If France and Austria cooperated more during Italian Unification and/or the Crimean War they might have developed better relations. It is surpassing actually that France didn't try anything to stop Prussia from growing until OTL's Franco-Prussian war. After all it is both their interests to see a weaker Prussia. But by the time France acted it was too late to stop them.
As far as whether they could have beaten Prussia together, I think the earlier the conflict the greater chance they have. It would be best to have Russia stay out of it. Italy is likely to go in Prussia's side regardless.
In the best scenario I see France and Austria able to maintain a semi-status quo for a while. In the long run it might mean the survival of Austria as a whole. And I could see an independent Southern German Confederation (based around Bavaria Wuttemberg Baden) independent from whatever Germany Prussia is able to make.
During the mid 19th century, any sort of German War of Unification means a similar Italian War of Unification will occur concurrently. This is simply because Austria was in both baskets and any Germany not directed by Austria is therefore going to be fighting against Austria. An interesting dynamic, really, as the survival of one is almost entirely dependent upon the survival and success of the other.
Now, France was actually quite happy to see Prussia and Austria go to war because they thought it would be a quick war and Prussia would be on the losing end. When - after the first week was up - they saw the Prussians in full dominance, they were totally shocked. They expected somewhat that the Austrians could salvage the day, but were not quite prepared for the conflict. Simply put, they didn't really have a valid Casus Belli. Had they jumped, that might well have pulled Britain in worrying over maybe Napoleon III taking up his Uncle's cause.
Of course, Bismarck was terribly worried about this which is why in the negotiations Prussia went so lightly on Austria. It is also why, when going to war with France, he made damn sure to court the powers and show them that it was
France who declared war, it was
France who was the aggressor over something simple, and Prussia was only defending herself.
1. Could it happen?
2. Under what circumstances?
3. Likely Casus belli?
4. What would the out come be in the event of war?
5. Best battle plan for each side?
6. The rest of Europe's reaction? Who takes who's side and who stays neutral who sends a strongly worded letter?
1: Yes.
2: Various options.
3: Alliance system seems pretty likely.
4: Depends, is Prussia alone? Has she thrashed Austria and formed the North German Confederation? What is her relationship with the south german states?
5: France - Dead rush into Prussia; Austria - consolidation and thrust through south german states while guarding the Sudeten Mounts; Prussia - draw one or another army into a delaying action, while crushing the other (likely kill of Austria quicker, as she's weaker of the two), then pivot to deal with the other.
6: Again, depends on the casus belli and whatnot. Britain may well stay out of it if its Prussia who got the ball rolling. Russia may come to Prussia's aid, depending on whose on the thrones at the time and their personal relationship... of course, it might also just come in to smack Austria a bit Italy will jump Austria, for sure, maybe France as well - or wait for France to jump them.
Very likely scenario, if the cards fall the right way.