I'm not sure there is any "typical" answer here. Spitfires and Hurricanes armed with .303 Brownings fired at 20 rps each gun, 160 RPS overall for 8 guns, for approximately 16 seconds.

Most 109s had twin MG-17s firing 17 RPS through the prop, and carried a minute of ammunition.

More here.

The 'number of rounds' carried varied wildly, especially when comparing planes armed with machine guns - firing similar rounds to the rifles carried by the troops on the ground - to planes armed with autocannons - firing rounds that might be used on the ground in an anti-tank rifle.

So while you might have many thousands of rounds of machine gun ammunition, you might only have around 100 rounds of autocannon ammunition.

Either way, the total firing time of the various guns would be measured in seconds - definitely a lot less than is depicted in (most) films and (most) games.

Edit: to avoid being misleading - of course a lot of WW2 fighters carried a mix of cannons and machine guns.

To add to what other people said, different countries often had different operational expectations for their aircraft. The two extremes are Russia and America. Russian developed aircraft were made for short range, low altitude interception/escort missions. Most Russian aircraft of WW2 used the ShVak 20mm cannon which fired at a rate of 700-800 rounds per minute (slightly more than the typical 600-750 rounds of other 20mm cannons). To put that into perspective, many early war Russian fighters were armed with a 20mm cannon that held 120 rounds and a heavy machine gun with only 200 rounds (the extreme is the Yakovlev-9t that only carried 30 37mm cannon rounds and a single HMG). American fighter aircraft were armed with multiple .50 caliber browning heavy machine guns which fired at about 800 rounds per minute and held about 400 rounds per gun, with up to 8 guns on a plane. American fighter aircraft were built for long range bomber escort missions rather than the frontline fighters of the Russians.

You also have to keep in mind that larger sized guns could do the job of smaller weapons using much less ammunition. Light rifle caliber machine guns like the 7.7 Brownings of the British were "weak" against other planes compared to explosive 20mm rounds. However, more 7.7 MGs could be fitted on a plane with more ammunition.

Wikipedia is an easily accessible source for individual airplane statistics if you look at the specifications section of each plane.

I'll just list off some of the better known fighter aircraft of WW2

American P-51 Mustang D variant, 6 .50 caliber MGs w/ 1880 rounds total

Mid war variant British Spitfire VB. 2x20mm Hispanos w/ 120 rounds total in addition to 4x7.7mm Browning w/ 1400 rounds total

Late war variant British Spitfire Mk XIV. 2x20mm Hispanos w/ 240 rounds total in addition to 2x.12.7mm Browning MGs w/ 500 rounds total. usedThe development of a reliable belt feeding mechanism for the Hispanos greatly increased the amount of ammunition that could be carried over the 60 round drums earlier planes

Early variant German Bf-109 E4. 2x20mm MG/FF 120 rounds total and 2x7.92mm MG17 2000 rounds total. Earlier variants had 2 additional MG17s instead of the 20mm cannons.

Late war German Bf-109 G6. Typically has 2x13mm MGs 600 rounds total 1x20mm MG151 200 rounds total. Could replace the MG151 with a 30mm Mk108 cannon with 65 rounds. Could also add 2 20mm gunpods under the wings.

German Messerschmitt 262 Jet fighter 4x30mm 360 rounds total

this is a particularly difficult question to answer given a) the number of different fighters and b) different doctrines regarding usage of fighters in offensive and defensive modes.

A Zero carried 60 rounds of 20mm rounds on each wing and roughly 1200 7.7mm machine gun ammo. Typically 7.7 is not very good against other flying machines so the 120 rounds of 20mm were used in dogfights. resulting in a fighter that while extremely manoeuvrable and agile did not have the ability to last long in a fight if up against skilled pilots.

As everyone else mentioned, "how much did they carry" varries wildly by plane and by weapon mounted on them.

For your other questions:

What was the affect of this on the length of dogfights?

If you ran out of bullets, you went home. Or you rammed your plane into the enemy^(1). Or, if you were engaging an enemy fighter in a dogfight, you would fight to keep your positional advantage until your wingman needed you to get out of the way so he could shoot. Without ammunition a fighter plane is helpless, but depending on the situation the opponent might not know that, and you could still serve a bit of purpose in spotting for your wingmen and giving a general impression of strength. So it somewhat depends on the mission. Patrol or interdiction? You landed and reloaded. Bomber escort? You likely stayed put to project imaginary power.

^(1)"Theo, I have run out of ammunition. I'm going to ram this one. Good bye. We'll see each other in Valhalla."

Also did Pilots who were on bomber protection mission conserve ammunition so that they could protect the bombers for the duration of the flight?

Pilots always conserved ammunition. Most dogfight time was spent angling for a positional advantage where the pilot could safely line up his shot and the target, and would pull the trigger for about 1 second. If he wasn't sure he was going to hit the target, he simply should not be shooting at it.

In video games you'll happily take hail-Mary shots at ranges upwards of a kilometer because you might land a hit, ammo is only a few moments away, and death is largely meaningless. In real combat, running out of ammunition could very easily be a death sentence. Not only for you, but also for the bombers you were supposed to be protecting, for your squad, and whoever your bombing raid was supposed to be supporting.