The situation was more or less the following.
Until 1936, France sees Italy as an ally, but not one useful enough to bother help in the League of Nations. Also, Germany remains the old foe, but is still useful to keep the Soviet Union at bay, and as long as they don't attack anything important or become dangerous will let them do as they wish (partly because they're still behind when it comes to rearmament).
Italy is neutrally hostile to Germany, as Mussolini considers Hitler a madman, and relatively friendly to France. France not helping with the embargo ordered by the League of Nations after the conquest of Abyssinia in 1936 and Germany giving the League the middle finger and helping economically Italy reversed the situation, with France seen as traitors and Germany as a necessary ally (by march 1938 Italy will throw Austria to the wolves). Note: at this point in time, Italy is the second strongest military power of continental Europe, as Germany is well behind in the rearmament process and France has just begun, while Italy has kept had never truly disarmed (Soviet Union is the biggest power, if only due sheer numbers). By 1938 both France and Germany will have surpassed Italy in terms of military power, as Italian industry was still weak and unable to fully support the needed modernization (the Royal Italian Navy will remain a power to be reckoned with, but the Air Force and the Army will be plagued by antiquated equipment, with the Army further weakened by the endemic corruption and nepotism of the officer corps and, outside a few specialized units and the Bersaglieri assault troops, truly capable only in mountain warfare).
Germany is hostile to France and friendly to Italy, but is still militarily weak (needing the Czechoslovakian weapon factories). Economic help and the renounce to former Austrian territories will gain Italy as an ally, and by 1938 there will be no problem at invading.
After the attempted coup in July, 1934, Italian troops were mobilized at the borders, but did not enter Austria. January 1935, Mussolini gave Hitler green light to annect Austria. At no point Italian troops were in Austria.
Checked. Mussolini didn't give the green light until 1938, but I have to concede that Mussolini never sent troops due the clear opposition of the Austrian public opinion (until they were invaded, at least).
And if Germany and Italy were to start a mad race into Austria, Germany would occupy most of the important territory by the time the Italians would have crossed the Brenner. The armies would probably meet somewhere near Graz.
By this point, German mechanized forces were badly organized and poorly coordinated (a factor unknown to everyone until the Anschluss itself), and the Austrian Army would have been able to stop them at the border.
The Brenner serves as a bottleneck for reenforcements. This would almost ensure a victory for Germany on Austrian territory, while doing the same for Italy if Germany crosses.
Assuming the Germans can pass the border, it's true.
Since Germany wants Austria, occupation is a win. (although Südtirol and Trentino would certainly fall to the Italians) Since Italy wants a free Austria, bombing it to rubble won't get them brownie points. Also, they would be restricted to aerial combat, since both sides could effectively block the Brenner by airpower. It's a classic mexican standoff.
1)Trentino and South Tyrol (Alto Adige in Italy) were already Italian territory (and still are). If you meant the still Austrian part Tyrol by that, then you're right, it would most probably fall immediately.
2)Italy's main objective was to keep clear the way to Hungary (with which they had good economic relations, and Austria was the best way to trade). What they need to call it a win would be the mountains in the south (that are guaranteed to be occupied, as the Austrians would be busy at the German border), Vienna and anything east of it. It depends on how long the Austrians can stop the Germans and the Wehrmacht can reorganize its troops, but if the Austrians can hold long enough the Italians occupy everything they're interested into and will reinforce them.
3)Nope. You can't block a mountain pass with air power, not with the airplanes and bombs of the time. The only ways were to march there and place a garrison or to blow up the mountain over it (a good reason for why the Italian soldiers of World War I hated the Austro-Hungarian ones was that the latters were better at sneaking up and set the explosive to blow up mountains under your feet). Or, if the nearby mountains had the right places, placing artillery to shell anyone trying to pass. Airplanes of the times were too light and had inadequate bombs for the task, and any attempt would mean give the troops a good laugh as the bombers try and fail to bomb them in spite of the winds.
Long story short: before Abyssinia, the smart thing for Hitler would be to stay clear of Austria, for an invasion would be kicked back and the French could decide he's too much trouble to keep and invade on the western border; after Abyssinia and before late 1938, the Italians can still defend Austria successfully, but they could decide it's not worth it (and in real life Hitler made sure the Italians decided so beforehand); after late 1938 (when the Czechoslovakian factories have produced enough tanks to quickly overwhelm the Austrians at the border) Germany wins, occupying the important territories before the Italians can pass all the mountains and move troops there, at which point the aftermath depends on Mussolini deciding to either recognize the new status quo and let him grab the country (most probable), grab the mountains at the south (possibly as bargaining chip for the link to Hungary) or go to war (highly improbable: the Italians are undefeatable in the mountains, but the Germans are superior in the plains, there would be little gain in case of victory and the other defender of Austrian independence is France, with which Italy isn't in good relations).