What role, if any, did Japanese anti-aircraft fire play at MIdway?

The USN, and anyone else in the dive bombing business, could tell you that in those days once a plane started its dive, it was very difficult for AA to shoot down . . . mostly a fire direction and time on target problem that, with locally controlled fire such as the Japanese employed through most of the war, generally leads to a bad defense solution. In other words, if the dive bomber knows his business, he’s coming almost straight down on a target, hopefully lined up straight with the direction of travel (this reduces HIS deflection problem) which probably has – I hesitate to use the word “small” caliber, but what I mean is something in the 12 mm to 20 mm range - a small caliber defensive suite which cannot be elevated to an angle which would allow for effective return fire.

In the overall scheme, Japanese AA didn’t appear to be all that effective anywhere during the war. Using official USN statistics, compiled circa 1947, in the 1941-42 period, US carrier-based aircraft flew some 1976 sorties attacking targets, encountering AA on 1238, or 62.6% of attack sorties. In the 1238 attack sorties meeting AA, 113 planes suffered AA hits. 47 of those were lost and 66 returned damaged; thus 41.6% of aircraft hit byAA fire were lost. Overall, however, this works out to only 9.1% of attacking aircraft suffering AA hits, with only 3.8% shot down. And this was Japanese naval AA’s best annual effort; as the war progressed, their results were increasing less effective despite increases in numbers of guns, especially land-based, and resultant volume of AA fire.

More recent research (see Parshall & Tully “Shattered Sword”) indicates that Japanese AA fire at the Battle of Midway accounted for only but two US aircraft during the various attacks on the Japanese carriers. Of the 47 losses noted above in the USN post war compilation, 24 were counted to have occurred during the Battle of Midway.

If we now know that 22 AA losses ascribed in the 1947 as to Japanese AA fire at Midway were actually the result of some other cause (probably to CAP action), then that would mean that USN AA loses for 1941-1942 actually were not more than 25, or 27.4% of those receiving AA hits (reduced accordingly from 113 to 91). In turn, the percentage of attacking aircraft suffering AA hits, overall, reduces to 7.3% and only 2.0% shot down. Either that or there were more than a few planes lost somewhere away from the immediate area of the Kido Butai whose loss was the result of AA damage. A little something to think about.

Official statistics on US losses for the Battle of Midway period are: 261 carrier-based (including strike escorts) and 59 land-based action sorties attacking Japanese ships for a total of 320 attacking sorties, with a total of 24 aircraft (20 carrier-based and 4 land-based) lost to AA, or 7.5% of all attacking aircraft. Total action sorties for the time period were 387 for carrier-based aircraft and 101 for land based. Losses credited to Japanese aircraft against total carrier-based sorties were 42, or 10.9%. For total land-based sorties, losses credited to Japanese aircraft were 33 or 32.7%. Total losses credited to Japanese aircraft: 65 aircraft or 15.4% of all action sorties. Total carrier-based operational losses were 16 (4.1% of carrier sorties).

There's a reason you don't hear much about Japanese anti-aircraft fire from the carriers: It really wasn't much of a factor. By design.

The Japanese philosophy with their carriers was that they were what gamers call glass cannons: Designed around overwhelming offense, with little effort put into defense. They could pretty much obliterate enemy naval units with their air power, so their offensive air power was intended to be their best defense. Failing that, the job of actually defending the carriers themselves from close attack was largely intended to be given to their counter-air units for air attacks, and their escorting naval units for naval attacks.

They weren't even designed very well to take hits or repair themselves after taking them, presumably because if they did their air job right, nobody should be getting that close.

It should also be noted that American anti-air guns of the time were using far more advanced early mechanical analog computers to help with targeting, in some cases paired with radar range finding, so they were relatively way more effective.

The US Navy Rangekeeper and the Mark 38 GFCS had an edge over Imperial Japanese Navy systems in operability and flexibility. The US system allowing the plotting room team to quickly identify target motion changes and apply appropriate corrections.

...

In contrast to US radar aided system, the Japanese relied on averaging optical rangefinders, lacked gyros to sense the horizon, and required manual handling of follow-ups

I checked and the Akagi in particular had all of 6 anti-aircraft guns that could be used against dive bombers. They were mounted rather low on the hull, so they couldn't all be used in the same direction unless that direction was nearly straight up (at which point its almost too late). In general, they'd be doing well if they could bring 3 of them to bear.

The overall effect of this was that Japanese carrier-based anti-air fire was a pale shadow of what the Americans could expect to put up.

Two great answers have already been given about, respectively, why the Japanese AA fire was not so effective, and how in reality it behave in the battle of Midway.

I would like to add a third consideration, which is about the general situation of AA fire in Japanese fleet and not on carriers only. If you consider some other battles such as the Cape Engano in 1944 or some air attacks in the Philippine Sea or the Coral Sea, you'll see that while air power had in the end the upper hand agains ships, it required multiple and massive sorties to sink a significant number of ships.

In that sense, Midway is a singularity because the Japanese fleet suffered very heavy losses from a not-so-numerous number of airplanes. A major explanation for this situation, compared to other battles I mentionned above, is that in Midway Japanese carriers lacked the support from the escort ships, especially heavy cruisers, battleships and numerous destroyers. Those ships were engaged in protection of the troops convoy, in a separate group, and this fleet fell back after the first signs of opposition from the Americans (= land-based aircraft attacking the Japanese fleet).

If this move of the transport fleet is understandable, because it saved it from air attacks while Japanese carriers were supposed to reduce Midway's airfields to ashes, this deprived carriers from this surface ships escorts that could have added much more AA firepower.

So to answer this part of OP's question:

Suppose there had been meaningful anti-aircraft fire against the torpedo bombers

This could have been possible if the Japanese had kept most of their war ships grouped.

And:

Would that have freed up Japanese fighters for action at higher altitudes against dive bombers?

Probably not because the Japanese still lacked the (radar-based) tactical direction to understand correctly the threat of the dive bombers.